Maybe it's the repetition during memorization that allowed me to understand what the poets were doing with meter and enjambment.
I'm mostly interested in iambic pentameter and heptameter: Shakespeare, Spenser, Arthur Golding, George Chapman, Milton, Tennyson, etc. I can't say that I understand a lot of modern, unstructured poetry yet.
The use of language is usually more interesting than the meanings of the poems for me (with the exception of a few, like Wilfred Owen). Memorizing a lot of poems creates mental structures of rhythms and sounds that are kind of like templates that your thoughts can fill, and they echo in my head for a long time afterwards.
I have to sometimes force myself not to pick up a poetry book, because it's so interesting that it can distract me from other things. Before that, poetry didn't interest me or make much sense.