Technically, to paraphrase these lines from Eliot's "The Love Song of J. Alfred Prufrock," you don't really need to know what the lines mean, or to interpret them. A paraphrase is just a putting in your own words what the writer writes, not an interpretation of what the writer writes. I'll still explain the lines for you, though, and it does seem that your assignment is to somehow incorporate interpretation into your paraphrase.
The lines are an allusion to the sirens of Greek myth. The sirens would lure passing sailors down into their water caves with their songs, in the process putting them under a spell. Once the sailors were in their caves, the sirens would stop singing, break the spell, and thereby drown the sailors. The mermaids mentioned three stanzas earlier are an allusion to the sirens, also.
Some commentators interpret these lines as applying to the speaker daydreaming, then being awakened when someone talks to him.
Either way, literally, "drown" refers to drowning by water. Figuratively, it may refer to being awakened out of the reverie of a daydream.