Salzman is maybe the most complex character in the story: an unhappy man who wants to make people around him happy. We first see him sporting "an amiable manner curiously contrasted with mournful eyes" (2) and later he appears as "a skeleton with haunted eyes" (116). So clearly something's bothering the man—and this dude has some super-duper haunting eyes. We imagine them as Steve Buscemeyes.
Yet despite being haunted—and despite the fact that he doesn't seem to be very good at his job—he yearns to bring young people together:
"I have two fine people that they would be wonderful to be married, I am so happy that I talk too much." (129)
He's a good (and very talkative) man with a whole lot of feels, but he's not a bitter man who lets his pain consume him.
It turns out that this well of pain stems from "my baby, my Stella, she should burn in hell" (188). We don't get a lot of details, but Salzman's daughter is basically a wild and crazy gal, who doesn't intend to be a poor girl forever. This being the 1950s, this could be code for sleeping around with rich men… although Malamud leaves this (and much else in this story) ambiguous. In any case, because of her bad behavior Saltzman has disowned her.
This is clearly eating away at Salzman from the first time we see him, and likely has for some time before our little tale begins. It's an old story—children break their parents' hearts and vice versa—but it does paint Salzman's job in a more interesting light. You could infer from the evidence that Salzman is trying to bring other young people together as a way of atoning for his failure with Stella: that by making his own girl unhappy, he's devoting himself to trying to make as many other people around him happy.
That gets even more intriguing when Leo raises the possibility that "Salzman had planned it all to happen this way" (200). In other words, Salzman was hoping Leo would find his daughter's picture and fall for her, thus redeeming them both. Clever girl.
That puts Salzman's primary motivation in sharp focus: this guy wants to make everything a-okay again. The greatest thing he can do is to heal his daughter's wounds by introducing her to a nice boy who will save her. Also, this match allows him to help Leo, who has, in the course of the story, learned to strive for the very connection Salzman wants to provide him with.