I agree with Estelyn Telcontar; all the poems and songs reinterate just how different all the peoples of Middle-earth are.
Also, I think it's Tolkien's style, showing he's a poet, not only a reader of prose.
It also adds some diversity in the story: to make sure you're not reading huge long chapters of solid prose, Tolkien dots every chapter with at least a small poem.
I don't think it's so strange anyway. I know I'm always bursting into song....
FRODO OF THE NINE FINGERS AND THE RING OF DOOM!....sorry. Now whenever I get bored I always suggest to my friends: Wanna sing a Hobbit drinking song?!
