As I considered this question, another question arose: Why does LOTR mean so much to me?
First, I like it because it's an adventure story. Yet there are many "adventure" stories, and none of them are quite the same as LOTR, because LOTR isn't just an adventure story. It has more depth to it, the feeling that it's truly a history of things that once were. Much of this feeling is because Middle Earth was so extensively developed by Tolkien. It has its own legends, songs, sayings and idioms. It has a rich and varied history, and LOTR is but a part of the story.
One reason I relate so strongly to this story is because of the characters. They are so down to earth and fallible, yet they display so many virtues we all desire: selflessness, self-sacrifice, loyalty, compassion, humility, and brotherly love. Another thing is the idea that you don't have to be great or powerful to make a difference. The characters weren't trying to be heroes, looking for adventure, fame or honor. They just did their duties, and ultimately, they did receive those things - but that wasn't the reason they did them.
So for me, LOTR is an inspiration to do my tasks cheerfully, whether they are mundane or perilous; an inspiration to try to cultivate the virtues that endear the characters to us all.