To the OP:
I usually either really like a poem, or have no patience with it; there is rarely a middle ground. I have not encountered much poetry that I like, but it is out there, and it can affect me on levels that more direct and objective writing cannot. Yes, there is probably emotional appeal involved, and I am sure that is a significant part of the impact. Most of my own writing, speaking, and other interaction tends to be very objective with limited emotional content. A good* poem shows me "the other side", what I am missing most of the time. A good poem gives me a different perspective on something, or causes me to notice and consider something I usually overlook.
I also appreciate the precision of language in a good poem. I try always to be precise in my own use of language, taking great care to use those words that will convey my meaning most exactly. Though poetry is usually subjective and non-scientific, it also requires a certain precision of language to convey the thoughts and images of the author. Good poetry has an economy of language, an efficiency in which every word must contribute to the whole; none wasted, or extra, or misplaced.
*"good" here is my own personal, subjective definition; in other words, poetry that appeals to me