How and why is "The Love Sone of J. Alfred Prufrock" a "love song"? I have read this poem several times before, but my professor asked us to figure...

In the case of this particular piece of literature, it is a poem. Poems used to be sung quite often in history, so one reason this would be appropriately titled is because it is simply that: a poem and poems used to be sung. 

Also, because it contains the thoughts of a single speaker, it is considered lyric poetry.  Lyric poetry was often sung.  In this day and age, we typcially associate lyrics with songs.  Lyrics are the words to songs (our modern-day definition).  Lyric poetry, however, is NOT necessarily sung, but it contains the thoughts and feelings of a single speaker.  Such is the case for this poem.

Finally, this could be a love song because it is about the misfortunes in love of the speaker of the poem, Alfred J. Prufrock.  Prufrock is bemoaning his lack of luck in love.  He is extremely shy and awkward with women and does not know how to best approach them.  He cannot overcome his own fears of rejection, etc., to take the risk in approaching them, although he dreams of doing so.  Love songs, as we all know and as the last editor pointed out, are not always happy, and this "love song" is one of these sad love songs.