You should play this game if you want to know the full story of Kingdom Hearts. But if you aren't a fan, don't bother.
Kingdom Hearts: Re: Chain of Memories is exactly what it's advertised to be: a game that links Kingdom Hearts and Kingdom Hearts 2. You will probably end up enjoying the storyline but despising the gameplay by the end of this and that is to be expected. Let me touch on a few details of this game to give you a taste of what to expect.
Storyline: The storyline literally picks up exactly where Kingdom Hearts left off with Sora, Goofy and Donald looking for their friends when they run into a strange, hooded figure. They follow him into a mysterious castle where Hooded Man 1 basically tells him the rules of the castle. They have forgotten all skills and spells and must fight with cards. Also, with each floor, they must use a card to venture through a world they have been to in the past. Finally, Hooded Man 1 tells Sora that he may find something dear to him in this castle. Something that was forgotten in the past. Spurred on by this riddle, Sora and friends decide to brave the Castle and find their friends.
Overall, the storyline is pretty good and does a great job of linking KH to KH2. I highly recommend you play KH before playing this game and I also highly recommend you play this game, or, at the very least, read a script that you can find on Gamefaqs, before playing KH2.
Another nice thing Re: Chain of Memories does for you is add a theatre mode that will be unlocked after beating the game. This will allow you to watch all the movies over again. Basically what this means is, if you want to play KH2 without playing this game, find a friend who has beaten this one and watch all the movies that way. It probably won't have the same emotion and effect on you that you would get playing the game through but you will certainly not be lost when KH2 starts up.
Gameplay: This is the section that differentiates KH: Chain of Memories from the other entries in the series. Chain of Memories is fought with a strange card battle/real-time fighting system that both works well and fails miserably. In the first half of the game, you will be able to create your own deck made out of cards you receive from a variety of places. These cards are all the attacks and items you used in the original game plus a few added extra to give you some hints to Kingdom Hearts 2. You will fight the same heartless as before but this time, you will have to use your created deck to fight, in real-time, by selecting cards and mashing the X button. For the most part, standard heartless battles are simple and you can pretty much fight the same way you did in the first game (provided your deck has enough keyblade cards in it) but boss battles are a different beast. Some will allow you to button mash while others require constant strategy. Overall, this leads to a strange disjointed feeling in the difficulty. While some bosses are incredibly easy, others are nigh-impossible to beat without help.
If this all sounds confusing, it's because it is. To top it off, the in-game tutorial is an absolute mess. While it teaches you certain aspects of the game, it completely leaves out other aspects that can lead to a completely faulty deck later in the game. I HIGHLY RECOMMEND YOU READ THE INSTRUCTION MANUAL CLOSELY BEFORE PLAYING THIS GAME. If you don't, you may find yourself in front of a boss that you will not be able to beat. One of the main criticisms of this game is that the sense of exploration in Chain of Memories is completely gone. I would totally agree with this. It will definitely leave you longing for Kingdom Hearts 2 long before the game is finished.
Sound/Graphics: The graphics are exactly the same as from Kingdom Hearts which isn't really a bad thing. KH had some nice graphics for it's time and while it looks slightly outdated now, consider this: they remade this game from the Gameboy Advance. I'd say what they did with it is pretty impressive.
As for the sound, the voice-overs are mostly the same as from Kingdom Hearts but for a couple of changes, some quite jarring, others almost unnoticeable. First of all, Hayden Panettiere is no longer Kairi. Instead they gave the part to Alyson Stone (sorry if I misspell any names, I don't have them in front of me and am trying to go from memory). This would be a big deal except for the fact that Alyson Stone does a great job as Kairi and sounds pretty similar to the original. One of the most noticeable changes in the voice-overs isn't from a change in cast but a change in age. Since Haley Joel Osment isn't exactly a kid anymore, Sora suddenly sounds all grown up. It's a bit shocking especially if you just finished the original. Riku is the only main character who made the transition completely unscathed. David Gallagher reprises his role and does a great job at it, definitely.
Hopefully, this review gives you a small idea of what Chain of Memories is like. If you get nothing else out of this please remember to read the instruction manual! Especially the parts about Premium cards. You may end up dying repeatedly if you don't. Good luck and enjoy!