As Erik P. said it is not necessary to know the story to play the game. I would add that it can be a good way to be introduced to the Lord of the Rings world.
I think the games goes quite deep into the story, but not in a way that gives an advantage to who has read the books. There are 4 maps: Moria, Helm's Deep, shelob's lair and Mordor (plus Bree and Isengard with the expansion Friends & Foes). Each takes you through an episode of the book. You'll face many events, gather shields, hearts, rings and suns, meet some famous characters - like Tom Bombadil, Glorfindel, Aragorn, Gandalf or Gollum - and call for Gandalf's help many times.
If you know the books (or even the movies) you'll know who these characters and places are. In the case of Gollum, that means you know that he's neither a friend nor totally a foe. This is made clear in the game as the card for Gollum has three stars and one die (dies are dangerous), so you won't be wronged by this lack of knowledge.
So while knowing the story will give you the reasons for why this event happens there and why this character has these specifications, the game will introduce newcomers fairly well.
In the end, even if you know the books very well, you'll be nothing more than a bunch of hobbits facing too great a destiny. In my case, this meant meeting Sauron in the Moria, many, many times...
Beside, Howe's illustrations are awesome!