Miller names this female character "The Woman," suggesting she has played an important role in Willy's past. His conversation with her is both revealing and significant. It shows that Willy is unfaithful to Linda in an ongoing relationship with The Woman; they meet regularly when he is in Boston on sales trips. He buys her stockings and entertains her by being funny and charming. This particular bit of dialog implies a great deal. When Willy tells her he will see her again the next time he is in Boston, The Woman replies, "I'll put you right through to the buyers." The implication is that Willy is pursuing a relationship with her to gain an advantage over the other salesmen. This indicates both his desperation and his lack of integrity.
Another exchange between them is also significant. When The Woman says that she picked him, after "watching all the salesmen go by, day in, day out," Willy is pleased. The idea is so pleasing to him that he asks her a few moments later, "You picked me, heh?" Willy is hungry for compliments and approval. Willy's feeling desirable, even to this woman about whom he cares nothing, indicates his insecurity.
In Arthur Miller's play, Death of a Salesman, the exchange with the Woman is significant because it shows that Willy has had an affair with a secretary in Boston. This scene does not indicate if the affair is still going on, but the audience learns that Willy brings her gifts and she find him funny. His ego is further bolstered by her pronouncement that she chose him of all the other salesman in and out of her office. Willy definitely appreciates the feeling, as he has just finished telling his wife how lonely it gets on the road when business isn't good.
It is significant that whereas Linda is truly dedicated to her husband, and in her eyes he is a successful father, husband and salesman, Willy lies to Linda by committing adultery, and lies also when he says he is lonely. It seems he then must also be lying when he says,
On the road—on the road I want to grab you sometimes and just kiss the life outa you.
This may be how he is feeling when he says it at home, but on the road, he obviously would not be concerned about Linda when he's busy with someone else. This also shows us that the relationship between Willy and Linda has not always been good. It would also appear that the Woman represents another example of appearance vs reality. While all appears well between Linda and Willy, as Linda's voice overlaps with the Woman's voice, we find that Willy has not been faithful. And while Linda and Happy may not know of Willy's indiscretion, we will learn that Biff does, which simply complicates the family dynamics as the play progresses.