How does poetry work in sign language?

Um, most people read poetry.

I don’t think I’ve ever heard poetry spoken.

To me (a Sign Language interpreter) poetry is one of the hardest things to translate, because poems can be interpreted in many different ways. As such it’s very much up to the interpreter to decide how to translate it. In any case, to translate poems you must be well prepared.
Phonetic rhymes doesn’t have the same effect in Sign Languages as it does in spoken languages. But signs can rhyme, if they look alike. There is a Danish children’s book in Sign Language, called “Blæksprutter Prutter” which means “Squids Fart”, and those two signs rhyme in (Danish) Sign Language. The book has rhymes such as “Mom and Dad says it’s forbidden to pee on the carrots”. All signs in that sentence are made with the index finger, that’s why it rhymes.
As for poetry for adults it’s more like making the signs ‘flow’ beautifully and tell a story like that, always very visual. I can’t write poetry in Sign Language myself, but I have seen some and I’m always impressed.
Translating poetry in a Sign Language to a spoken language will never give you the same experience, just like watching a spoken poem translated to Sign Language won’t be the same experience, in my opinion.
It’s all very hard to explain, but I hope you get some idea how it works.