This guy is a real life version of McGruff the Crime Dog. He works for the Waterfront Crime Commission, trying to take down union gangsters like Johnny Friendly. Glover things he can work on Terry Malloy and get him to talk, sensing he might be harboring feelings of guilt…
At first, Terry tells him to scram. But, later, Glover gets under his skin. He says he remembers seeing Terry lose the biggest fight of his life. Terry's wounded by this, and explains that he threw the match because his brother wanted him to.
It's a clever tactic on Glover's part. By making Terry think about this stuff, he demonstrates that Charley and Johnny don't really have Terry's interests at heart. They're out for themselves, like they've always been. It helps Terry see what the moral option is: to squeal.
Glover and McGruff will probably toast each other over this later on.