The reason that the children were brought to Narnia is found near the very end of VDT.

"Dearest," said Aslan very gently, "you and your brother will never come back to Narnia."
"Oh, Aslan!!" said Edmund and Lucy both together in despairing voices.
"You are too old, children," said Aslan, "and you must begin to come close to your own world now."
"It isn't Narnia, you know," sobbed Lucy. "It's you. We shan't meet you there. And how can we live, never meeting you?"
"But you shall meet me, dear one," said Aslan.
"Are--are you there too, Sir?" said Edmund.
"I am," said Aslan. "But there I have another name. You must learn to know me by that name. This was the very reason why you were brought into Narnia, that by knowing me here for a little, you may know me better there." VDT, The Very End of the World

The children are brought to Narnia so that they can know Aslan better.

As for their remembering, the movies got it wrong. In the books their experiences in Narnia fade like a dream. Just like their time in England fades like a dream while they are in Narnia. The longer it is, the more dream-like it becomes.

So they lived in great joy and if ever they remembered their life in this world it was only as one remembers a dream. LWW, The Hunting of the White Stag

"I wonder, was it really the hall," said Susan. "What is that terrace kind of thing?"
"Why, you silly," said Peter (who had become strangely excited), "don't you see? That was the dais where the High Table was, where the King and the great lords sat. Anyone would think you had forgotten that we ourselves were once Kings and Queens and sat on a dais just like that, in our great hall."
"In our castle of Cair Paravel," continued Susan in a dreamy and rather singsong voice, "at the mouth of the great river of Narnia. How could I forget?"
"How it all comes back!" said Lucy. PC, The Ancient Treasure House

It was like meeting very old friends. If you had been there you would have heard them saying things like, "Oh look! Our coronation rings--do you remember first wearing this?--Why, this it the little brooch we all thought was lost--I say, isn't that the armor you wore in the great tournament in the Lone Islands?--do you remember the dwarf making that for me?--do you remember drinking out of that horn?--do you remember, do you remember?" PC, The Ancient Treasure House

"Oh bother, bother, bother," said Susan after she had thought for a moment. "I remember now. I took it with me the last day of all, the day we went hunting the White Stag. It must have got lost when we blundered back into that other place--England, I mean." PC, The Ancient Treasure House

In a moment she had bent the bow and then she gave one little pluck to the string. It twanged: a chirruping twang that vibrated through the whole room. And that one small noise brought back the old days to the children's minds more than anything that had happened yet. All the battles and hunts and feasts came rushing into their heads together. PC, The Ancient Treasure House

"But, Peter," said Lucy, "look here. I know I can't swim for nuts at home--in England, I mean. But couldn't we all swim long ago--if it was long ago--when we were kings and queens in Narnia? We could ride then too, and do all sorts of things. Don't you think--"
"Ah, but we were sort of grown-up then," said Peter. "We reigned for years and years and learned to do things. Aren't we just back at our proper ages again now?" PC, The Dwarf

I don't think Edmund would have had a chance if he had fought Trumpkin twenty-four hours earlier. But the air of Narnia had been working upon them ever since they arrived on the island, and all of his old battles came back to him, and his arms and fingers remembered their old skill. He was King Edmund once more. PC, How They Left the Island

Even Lucy was by now, so to speak, only one-third of a little girl going to boarding school for the first time and two-thirds of Queen Lucy of Narnia. PC, The Return of the Lion

That was probably a quote overload. Anyway, the above quotes show how the children change as they travel back and forth. When going from Narnia to England, they lose their abilities, their experiences, even their full memories of Cair Paravel. When going from England to Narnia, those abilities, experiences, and memories come back and what they have gained in England fades. Whichever way they go, they know they were in the other place, but the specifics are more in the subconscious. Because they are in the subconscious, they do not have the negative effect that the movies and most of fanfiction apply. They of course want to go back, but they aren't struggling, angry teenagers who can't cope with life as a schoolkid (I'm not fond of the way the movies portrayed the Pevensies, can you tell? ).

And their time in Narnia makes them work harder at some skills in England.

Lucy thanked her stars that she had worked hard at her swimming last summer term. VDT, The Picture in the Bedroom When she couldn't swim for nuts at home during their last adventure.

"Hast any skill with the bow, maiden?" said Tirian.
"Nothing worth talking of," said Jill, blushing. "Scrubb's not bad."
"Don't you believe her, Sire," said Eustace. "We've both been practicing archery ever since we got back from Narnia last time, and she's about as good as me now. Not that either of us is much." LB, How Help Came to the King

The movies also got wrong what it means to be a king in Narnia.

Aslan asks Frank a series of questions when Frank is unsure if he is qualified to be a king. "Can you use a spade and a plow and raise food out of the earth? ... Can you rule these creatures kindly and fairly, remembering that they are not slaves like the dumb beasts of the world you were born in, but Talking Beasts and free subjects? ... And would you bring up your children and grandchildren to do the same? ... And you wouldn't have favorites either among your own children or among the other creatures or let any hold another or use it hardly? ... And if enemies came against the land (for enemies will arise) and there was war, would you be the first in the charge and the last in the retreat? ... Then...you will have done all that a King should do." MN, Digory and His Uncle Are Both in Trouble

"For this is what it means to be a king: to be first in every desperate attack and last in every desperate retreat, and when there's hunger in the land (as must be now and then in bad years) to wear finer clothes and laugh louder over a scantier meal than any man in your land." King Lune, HHB, Rabadash the Ridiculous

Peter as High King should have known both what Aslan told Frank and what Lune told Cor. His time away from Narnia should not have made him power-hungry because being a King has nothing to do with power. Yes, he ruled Narnia, but that doesn't mean he lorded it over Narnia.

And if read all that and it didn't make sense please tell me. Sometimes what I think out in my head doesn't make much sense to anyone else.

I think that was one awesome post, and you get the point across very well. But I think there is a couple more that can be made. Just as the children worked harder in the real world at skills they could use in Narnia, their Narnia adventures were opportunities to learn vital things for real life.

I agree that the memory of Narnia would fade in real life. It explains why Susan would get so caught up in her life that she would forget Narnia. But why didn't the others forget Narnia also? Or the lessons about life they learned there?

Peter as High King should have known both what Aslan told Frank and what Lune told Cor. His time away from Narnia should not have made him power-hungry because being a King has nothing to do with power. Yes, he ruled Narnia, but that doesn't mean he lorded it over Narnia.

Actually Peter was away fighting giants when the events of HHB took place. One gets the feeling that as High King fighting giants he was kept rather busy, and that the other three were deputising for him quite a bit. In fact at least some of the triumphs he had, such as the pursuit of fair government, were in his name rather than something he did personally.

Something he appeared to have forgotten when in PC (book as well as film) he failed to believe Lucy had seen Aslan. Movie Peter shows he is still fighting giants in PC in his mind and hasn't adapted to changed circumstances as he should. That there were things about Narnia and himself he never knew even as High King. He can't rest on past triumphs, he has to learn when to move on and to learn when to accept direction. And so in both book and film Peter has lost his way.

In VDT he is in the unenviable position of studying for final examinations, another way of seeing how he continues to 'fight giants', this time with Professor Kirk's directions. It is also something Peter would have had to do regardless of what the rest of the family has to do if he wants to prepare for the rest of his life. And until LB we never hear from him again.

What Lucy and Edmund, as well as readers of the books, need to realise that nobody is perfect, and that all sin, even book characters. Not only Eustace. Lucy and Edmund have to be reminded that money and beauty aren't everything in life. That is what makes the books satisfying, knowing that the heroes are just as flawed as real life people. And I suspect that the real lesson of Narnia is how to learn to rule oneself.

Just as the children worked harder in the real world at skills they could use in Narnia, their Narnia adventures were opportunities to learn vital things for real life.

Thank you for making this point. I was going to get to that but time ran out. Yes, the children learn vital life-lessons while in Narnia. The things they learned in Narnia came back with them. Just think of Eustace's change between VDT and SC. He goes from being on of the bullies "hangers-on and tale bearers" to keeping the secret about Spivvens and standing up to Carter about the rabbit.

Actually Peter was away fighting giants when the events of HHB took place. One gets the feeling that as High King fighting giants he was kept rather busy, and that the other three were deputising for him quite a bit. In fact at least some of the triumphs he had, such as the pursuit of fair government, were in his name rather than something he did personally.

I guess I didn't explain that point fully. Yes, Peter was fighting the giants at the time of HHB. No, he was not present when King Lune told Cor what it means to be a king. However, as a king he would have known what Lune meant by it and what Aslan meant when He talked to Frank. Peter is not perfect, but he knows what it means to be a king.

I agree that he loses his way in the book as well, but it's not quite the same as in the movie. It's all in the tone. Book Peter seems insecure because he can't find his way in the land he once ruled. Movie Peter seems insecure because if he follows Lucy he won't be in the lead.

Absolutely. It awakens the fantasy and imagination inside of them. It gives them new knowledge, insight, and experiences. Who knows what going to Narnia may have inspired the kids to do that is simply not said in the books?

Now that you mention it, I would tend to let it be thought of as summer camp. Youth groups get togeter, go away somewhere, have real God encounters, and then get thrust back into the world. I always hated the last day of camp because it meant we had to go back to our ordinary lives. Fortunately it was always made a point that what we had there at camp wouldn't leave us; it would shine out as brightly as we ourselves would allow.
Aslan meant for them to keep themselves, keep up the spirit and the loyalty to Him even in the ordinary world. That's what Narnia is about, that's what Narnia is.

daughter of the King, I like your explanation!

Thanks for the reply, daughter of the King! Your post did make sense. I hadn't thought about it that way before, but you are right, the memories of Narnia are more like memories of dreams in the books. And I agree with Wagga, that would help explain how Susan could possibly forget about Narnia and abandon it for the lifestyle she chose on Earth.