This scene takes place in a room at the Plaza Hotel. Nick has just driven into town with Jordan and Tom. On the way they had to make a stop at Wilson's garage in order to get gas. While there, Wilson asks Tom again if he can buy his car because he needs to make some money so he can take Myrtle away from there. He tells Tom he ". . . just got wised up to something funny the last two days." Tom realizes, of course, that Wilson has discovered Myrtle is having an affair. While Myrtle has tried to be discreet and not alert Wilson to her adultery, Tom has made no such attempt to keep the affair quiet in front of Daisy.
Up to this point Tom has suspected that Daisy and Gatsby are having an affair, but it is blatantly clear to Tom when he sees Daisy and Gatsby together. This direct knowledge upsets Tom, especially since he feels Gatsby is so far beneath him socially. Tome says, "I suppose the latest thing is to sit back and let Mr. Nobody from Nowhere make love to your wife." Then he makes the comment about people sneering at family institutions. The irony here is that he has been making love to Wilson's wife, but he doesn't see the problem there. Tom only gets upset when another man steps into his territory, especially since it is Gatsby. Tom doesn't see his own hypocrisy here but Nick does because he's watched Tom and his affair with Myrtle.
Here is the quote from Nick:
Angry as I was, as we all were, I was tempted to laugh whenever he opened his mouth. The transition from libertine to prig was so complete.
The important words here are "libertine" and "prig". Tom was a man who felt at liberty to have an affair, insisting that it in no way comprised his family life. Then, suddenly, he is so traditional as to assert that any one who challenges the sanctity of family - as he has done by being unfaithful - is a destructive person. Suddenly for Tom, affairs aren't so funny.