Cleanth Brooks is New Criticism, a literary theory (some people prefer to call it a method of reading) that says, "Hey—everything you need is right here in these lines before you. Let's look at the beauty of the poem, not the political rage felt by the author when he wrote it. Focus on the text itself."
When people think of New Criticism, Cleanth is likely the first dude they think of. Let's cut to the chase: his project of close reading is thought of as the opposite of New Historicism (all the rage theses days), which asks questions like, "Hey, how was the stock market on the day this poem was composed? What was the poet's sex life like? Did indigestion produce that kooky rhyme scheme?"
Brooks's success as a New Critic has led some tormenters to accuse him of being irresponsible and not caring about the living reality outside of the poem. Huh? Well, these people like to point out that New Critics don't care that a real human author wrote the darn thing; it's like they think the poem wrote itself.
(We're not so sure that New Historicists care much about the author, either, but far be it from us to argue...)
Either way, Brooks's main commitment was to his reader. He was just part of that rare breed of decent critics who would do everything in his power to draw back the curtain on the irony, paradox, and ambiguity of poetic expression. Brooks wanted to make things clearer for readers; he wasn't about just taking complicated theories and tossing them into a blender with literature.
Best of all: no jargon. This guy's on your side. Students of poetry don't love Brooks because he wears a black turtleneck or has a French accent (seriously: he was from Kentucky); they love him because his books, such as Understanding Poetry, just straight-up taught people how to read poetry and get something meaningful out of those jumbles of words.