Hume's approach to philosophy was empiricist; he believed knowledge comes through sense experience, so in many ways he was a scientific thinker. Along with this he was also a skeptic and a naturalist, meaning that he did not entertain the idea that a transcendent world existed. His thought covers areas such as what is now called philosophy of mind, cognitive science, neuroscience, moral psychology, philosophy of religion, aesthetics, and ethics. His influence in all these areas is considerable, with experts in those fields who endorse his views or carry on with them called "Humean".
The reason I revere Hume could perhaps be considered twofold. First, I am very impressed with the man himself, I love his attitude and his style. He had the characteristics of a person that could be a good friend; a superior intellect, a wit, and a fierce loyalty to people. His willingness to defend his views even despite them costing him professorships and other appointments shows his dedication to the Socratic principle; following the evidence wherever it may lead. His sense of humility and disappointment that he expresses in places such as My Own Life show that while he did achieve fame during hi slide, he was disappointed that his work had been so misunderstood by his comrades in Scotland. Simply put, Hume was more than just a great philosopher; he was a great man.
Secondly his work itself. First, Hume is simply a joy to read, being called "the most important philosopher to write in English". He writes so well that even if you disagree with all of his ideas, you will enjoy having read him. His empirical approach to philosophy, his skepticism about his own findings, and some of the things his findings undermine also have had a great effect on me. For instance, Hume wrote in the Treatise that since knowledge comes from experience and we cannot observe ourselves, that the self is an illusion and we just have a bundle of perceptions about the self. His problem of induction is also one that has left me dumbstruck from time to time because while I sometimes wished he were wrong, I always end up concluding he was right.
By way of personal anecdote, before I read Hume I had questioned the idea of a finite God after reading Thomas Aquinas' Five Ways. However, after reading Hume's Dialogues and his argument that given the imperfections in the universe and our experience of design, it is more likely that several finite God's designed the planet rather than one infinite God. I had never thought of it that way, but after reading the argument and considering it, I came to the conclusion that Hume was right and remained in the LDS Church rather than joining the Catholic Church.
While I could go on and on about him, it will be sufficient to say that Hume has made me the man and thinker that I am today. Like Hume, I am a staunch empiricist, naturalist, and skeptic. This way of thinking has carried over to how I approach religion, politics, friendship, and obviously philosophy itself. I consider Hume, along with Jesus of Nazareth, Joseph Smith Jr,, Isaac Newton, and Charles Darwin to be among the 5 greatest men who have ever lived. I thank God daily that my friend introduced me to Hume and his thought; my life will never be the same because of it. God be thanked for David Hume.