I analyze the things I like constantly trying to determine why it is I enjoy them. I do this partly because I am a critical person by nature. Movies are what I understand more than anything, and so I'll stick to that for my example. Silence of the Lambs is well regarded and a lot of people give the movie tons of praise. They talk about the acting, the characters, the suspense generated by the pacing, but my favorite thing about the movie is nearly never discussed. The cinematography is freakin' superb!
By way of example, I always tell people to go back and watch the scene where Clarice is talking to the good Dr. in his new, lavish cage. What I noticed the first time I watched it was the way their back and forth evolved. As Clarice starts off with the upper hand in their discussion, the camera flips from her to the Dr. and we see him sitting there behind bars, caged and restrained. But as the scene evolves and the Dr. gains greater and greater leverage in their talk, the camera gets closer and closer to him, leaving shots of Clarice more distant, making her look smaller, weaker. Finally, as the Dr. seizes his position of power in their dialogue, the camera is in so close that we no longer see him through the bars of the cage. We are up close and personal with the monster, the audience themselves no longer protected by the view the camera was giving them. And when we flip back to Clarice? Now we see her through those bars, trapped and restrained.
That scene illustrates how very powerful something as easily ignored as camera work is. Just by changing a few things about the shot could have robbed it of its impact. As it was done, the cinematography doesn't just support the excellent performances by the actors, it takes the scene to places it would have never gone with just acting alone. This is why I think about what I like. So that I can gain real appreciation for what makes something good. If I had never taken the time to think about it, I'd never have realized what sort of impact cinematography can have on a scene and thus never value it like I should when I see it.