The narrator, Prufrock, repeats that the "women come and go / Talking of Michelangelo" because this emphasizes how many times he has been to a party just like this one, talking about the same subjects. His life goes in circles instead of getting to a destination. He is distressed with himself for frittering away his time at such another party instead of working on his art. It is numbing to keep going to these parties, where he says he measures out his life in coffee spoons. He feels like an insect pinned to a wall, unable to break free.

Interestingly, the speaker asks sixteen questions over the course of the poem, most of them around the theme of if he dares to "disturb the universe" by creating literature rather than simply attending events in literary and intellectual circles. He asks if he has "the strength to force the moment to its crisis."

He asks, too, if it would "have been worth while" to come back from the dead and bring a message for humankind, instead of going to these endless parties. However, he finds that

It is impossible to say just what I mean!

The speaker of "The Love Song of J. Alfred Prufrock” by T. S. Eliot tells us more than once that the women "come and go / Talking of Michelangelo" because he wants to convey to the reader that while the speaker/narrator is pondering life in a serious way, these women are, somewhat flippantly at a social gathering, lightheartedly moving from room to room and discussing art and may not be attuned to deeper, more important issues (or don’t care if they are attuned to them).

T. S. Eliot inserts the phrase, “In the room the women come and go / Talking of Michelangelo" at two critical junctures of the poem. The first is after the first English stanza. In the first stanza, the speaker talks of streets “that follow like a tedious argument," and also the evening spread out and likened to “a patient etherized upon a table." Therefore, it is apparent that the speaker is tired with life and even the evening about him makes him think of a person under ether on an operating table, possibly in a life-threatening situation. This sets the somewhat somber mood of this poem as the speaker, a man, thinks about his life—the past and his mistakes, the present and his inertia, and his less-than-enthusiastic view of the future. This is a counterpoint to the pleasant and cheerful evening the women are having.

The second use of the phrase is after the speaker talks about putting on a façade to ready himself to meet people in different situations. This shows the man’s misanthropic and cynical view of life and personal interactions and relationships.

The speaker also talks about there being time for “a hundred indecisions." As such, he is saying that he cannot commit to taking a stand or being resolute. It is as if he’s just rolling through life without enthusiasm and verve or the mindset to be purposeful and clear-sighted enough to make quality decisions. Therefore, he is somewhat like the women who "come and go” and engage in gentle banter as they enjoy a night out with somewhat light conversation. So, in one case the speaker is different than the women. In the other case he is a little like the women.

J. Alfred Prufrock longs to ask, essentially, “What is life all about?” In other words, “What is our purpose in all this theater or drama here on earth?” He looks at life and all its situations and relationships and strivings and wonders if it is all worth it in the end if one doesn’t have a strong purpose and a hope for the future.

The speaker wants to be bolder and more daring. He states that he is really the following type of person: “Deferential, glad to be of use, / Politic, cautious, and meticulous.” Consequently, he longs to ask the above-mentioned questions, as well as, “How can I live a less fearful life?”

In "The Love Song of J. Alfred Prufrock" by Eliot, why does Prufrock keep saying "the women come and go/talking of Michelangelo?"

The couplet in "The Love Song of J. Alfred Prufrock" really doesn't make any sense.

In the room the women come and go
Talking of Michelangelo

Imagine a whole roomful of women talking about Michelangelo, most probably in twos and threes, some of them exiting together in couples still talking about Michelangelo and then returning while still talking about Michelangelo as others exit in couples and threesomes talking about Michelangelo and then returning still talking about Michelangelo in order to join others who are talking about Michelangelo. What do they "go" for? If they have to find a restroom, do they continue talking about Michelangelo in the ladies' room and then in the hallway on the way back to the main room to talk about Michelangelo some more? Or do they stop talking about Michelangelo when they leave the room and resume talking about him when they return? If they stop talking about him outside the room, what do they talk about?

How can women come and go in a room? They can come into a room and go out of a room, but they can't do both, can they? And why so much coming and going?

The couplet sounds impressive--and it sounds like T. S. Eliot.

In "The Love Song of J. Alfred Prufrock" by Eliot, why does Prufrock keep saying "the women come and go/talking of Michelangelo?"

One really important piece of information to understand when reading "The Love Song of J. Alfred Prufrock" is that Eliot really loves to incorporate all sorts of allusions and references into his poetry: mythology, literature, philosophy, religion, history, art.  So when Eliot writes:

In the room the women come and go
Talking of Michelangelo. (Stanza II, lines 13-14.)

he is referring to the famous Renaissance painter and sculptor from Italy, Michelangelo.  Prufrock seems fixated on these women and their preference for Italian artists, but this line actually has very little impact to the overall meaning of the poem or Prufrock's question.  Eliot may have included these lines in the poem for several reasons.  Prufrock may be impressed by these women and their preference for high-brow art; perhaps he even feels intimidated by their culture and sophistication.  As the poem progresses, the reader begins to understand that Prufrock does seem to have a pretty low self-esteem when it comes to the opposite gender. 

In "The Love Song of J. Alfred Prufrock" what is implied in the observation, repeated twice, that women drop in and out, "Talking of Michelangelo" ?

The implication of this quote here is that the women at these parties are talking of trivial, non-important things.  They are discussing art, which today might be the equivelant of discussing the latest t.v. show or rock album that has come out.  He repeats this phrase to emphasize that at these tea-parties, the women and men do not discuss anything important, anything profound, anything serious, or anything that could be life-changing, distressing or deep.  He is frustrated with this; he feels that life is just a series of monotonous and meaningless conversation; he feels like he has "measured out [his] life with coffee spoons" instead of meaningful experiences.  So, the repetition is to emphasize the shallow nature of his social gatherings in contrast to the more serious experiences and discussions that he wants to have.

The repetition also emphasizes how the conversations at the parties are so shallow compared to whatever question he wants to ask the woman that is alluded to in the poem.  Whatever that question is (many believe it is a marriage proposal), it is big.  It is serious.  It is life-changing.  It is deep and profound.  He compares asking it to "rolling the universe into a ball", to "disturn[ing] the universe", to "murder[ing] and creat[ing]", to a "crisis".  So, take those descriptions, and compare it to the trivial discussions of "Michaelangelo".  No wonder he kind-of chickens out, and after asking, "would it have been worth it" to ask that question in such a setting, to such a woman, decides "No!".

I hope those thoughts help; it's a tricky poem at times.  Good luck!