Malamud falls back to the old stand-by of third-person omniscient to tell us his tale. He moves us effortlessly wherever we need to go, and doesn't feel bound by the perspective of one single person.
We largely stick with Leo throughout the story, sensing his feeling and self-recriminations, such as when,
He gradually realized—with an emptiness that seized him with six hands—that he had called in the broker to find him a bride because he was incapable of doing it himself. (110)
If we stayed with Leo throughout the whole story we'd be stuck with third-person limited, which shows us everything through one single character. But every now and then, we switch perspectives from Leo to Salzman, such as when the matchmaker notes:
… with pleasure the long, severe scholar's nose, brown eyes heavy with learning, sensitive yet ascetic lips, and a certain, almost hollow quality of the dark cheeks. (4)
Malamud wants us to know that Leo is our main dude, but doesn't feel bound to it when Salzman has something we really, really need to know.
In addition, third-person omniscient narrative lends the suggestion of a higher power; we're seeing what God sees and can flit back and forth between these characters without even asking permission. In fact, third-person omniscience is sometimes glibly referred to as a God's-eye view. For a story with so many religious overtones (and the sense of something greater than us moving behind the scenes) the technique fits like a glove.