Poprishchin says he's mad at the section chief because the section chief points out that Poprishchin is over forty and chasing the director's daughter even through he doesn't have a kopeck (that's Russian for penny) to his name or any plans for his life.
He rationalizes that he's only forty-two and can still move up the ranks to become a Colonel or something higher. All he needs are clothes of the latest fashion. There is a tiny problem, though: he's broke.
If you're starting to wonder why Poprishchin is so obsessed with rank and nobility, it's probably because he's very close to becoming a true nobleman. He's a Titular Councillor, which is rank 9 out of 14 in the Russian civil service of the day. This officially makes him a nobleman, but he can't pass his title along. If he moves up just one rank, to Collegiate Assessor, he'll be considered a nobleman who can pass on his position by birth. Being so close to real nobility could make anyone mad, really.