The simplest explanation is that they keep the peace. They are the military wing of the Capitol, dispatched to the Districts to maintain order, reinforce rules, and guard transports of both manufactured goods and raw materials back to the center of Panem.

[It's the kindest answer Effie can give. While the Peacekeepers are required to protect her, she has always found them a little unnerving. Maybe even a little scary, although she would never be able to explain why this is.]

It would be difficult to keep the Districts in line without them. May I enquire as to your name and reason for your interest in this subject?