Stream-of-consciousness writing is where the author jots down whatever comes into their head, in whatever order it comes.  It is usually centered around one main topic, but within the bounds of that topic, tends to wander about a bit.  It is a bit random, flowing, dream-like, passive, and non-committal.  Prufrock lays open his mind for us, and lets us see-without filters or organization-what he is thinking about.  It is a truly honest exposure of his thoughts.  Because he doesn't filter his thoughts, because he doesn't think twice before putting them down (which is what stream-of-consciousness is), we can understand him more honestly, and see him for who he truly is.  We see how paranoid he is about scrutiny, how insecure, how weak, how much of a coward, all of the excuses he makes, all of the going back and forth that he does.  If he had decided to write a poem not in stream-of-consciousness style, he would have written only the best arguments for not asking the woman his question; he would have hid and covered his weakness and insecurities.  But he doesn't in this poem, because of its style, and so we are able to see his mind wander all over the argument, and understand fully why he chickens out in the end.

Like I said, I encourage you to try to answer these questions yourself; use the eNotes study guides (I provide links at the end of each question), which are very helpful, along with your teacher.  I hope this helps!  Good luck!