The reasons for LOTR's greatness are fairly uninteresting; it's beautifully directed, wonderfully acted, expertly paced and masterfully written.
Lord of the Rings is essentially the template from which the majority of modern fantasy draws, weather it imitates that template or subverts it, there is a persistent unspoken assumption that everyone who watches or reads fantasy knows the basic story of Lord of the Rings.
Lord of the Rings has become the quintessential epic in the same way that Star Wars has become the quintessential hero's journey; they're the most primeval human narratives manifested in their purist form with great skill.
So I'm at a loss at how to sell you LOTR without going down the list of story telling 101's, but I'll try.
I suppose the first thing I would point out about Lord of the Rings is that it's immersive in a way that few films are; the opening is technically nothing but exposition, but every detail of middle earth is so rich and so well presented that you don't notice; LOTR has such a mastery of tone and is so wonderfully acted that it can communicate important details to you without saying a thing.
When you first see Bilbo with the ring, you know right away that it's bad news, when you first meet Aragorn, you know he's more than he says he is despite the fact that all he does is chastise the Hobbits about drawing attention to themselves and tell us about the Ring Wraiths.
This mastery of tone permeates the entirety of the story; you won't care how many times you think you've heard the story of good VS evil before, you have never seen it as clearly as you do know.
All of these things would make for a good ride, but not a profound story; that comes to us in the form of the stories primary theme; friendship and hope. These are two things that are so often thrown into otherwise bland stories to try and lend them weight, but LOTR understands how these things really work.
That tone I mentioned earlier is often very ominous; LOTR may be high fantasy but it is marred by none of the predictability that is typical of the genre. LOTR body count is actually very low, but you find yourself so attached to the characters, so involved in their struggle, that you can never escape the dread that even one of them might not make it to the end of the journey.
This is owed to two things; one is the acting, which is wonderful across the board, with Sir Ian McKellen as Ganbalf, Sean Bean as Boromir and Sean Astin as Samwise "the Brave" Gamgee being my personal favorites (The latter of which is also my favorite character).
The other is that all important tone I talked about; you can't help but find yourself attached to these characters whom you share so many adventures with. You become involved if only because they are the catalyst with which you experience the depth and quality of the setting. You attachment to them is compounded by the heartwarming courage and loyalty they show in the face of danger.
The characters in LOTR could be considered broad, but they are so sincere that you require almost no information about them to bond with them; just as the world is vivid enough to feel like a real place, the bonds the characters share feel genuine.
So I've described the way that LOTR engages you emotionally, and that is honestly it's primary form of engagement; you'll spend so much time in awe of the setting and dreading the fates of the characters that you won't have time to stop and think about what all this means; thankfully, the movie just tells you. LOTR does more that showcase the bonds of friendship and the need for hope, it allows you to experience them.
LOTR rings is a story about the strength and beauty of the human spirit; it admits that everyone is corruptible, and that evil is an ever present force, but it illustrates that within every human there is something fundamentally pure that darkness can't touch, that our constant struggle for survival is justified and worthwhile.
Lord of the Rings paints a picture of the very essence of hope, why we need and why it's worth having.
"There is some good in this world, and it's worth fighting for"
P.S: The Hobbit movies are okay, if you fall in love with the originals, you'll be happy just to get back to Middle Earth, but they have yet to even approach the quality of the original trilogy, though I personally never expected them to.
EDIT: By the by, if you think you have the attention span for it, see the extended versions; there is some stuff thrown in that doesn't need to be there, but there are some real gems to.
