So there's two different scenarios in which your system's volume could increase. Call #1 "balloon" and #2 "piston".
In a balloon, you have a volume that can respond to the conditions inside it. So if you heat the air in the balloon, it expands until the pressure is at equilibrium again. This expansion pushes the surrounding air out of the way, doing work.
Now, in a piston, you have a volume that you control directly, and the conditions inside it change in response. If you push down on the piston, thus reducing the volume, you haven't changed the kinetic energy of any of the particles, but you have made them closer to each other and reduced the surface area they have to bounce off of (inside the piston). The pressure in a system is the force by the contents on its containing walls. So in reducing the volume, you up the pressure.
The previous paragraph is about what happens at constant temperature: pressure changes inverse to volume. But sometimes we want to know about a system with constant pressure. The only way to make it so you don't up the pressure when you reduce the volume is to reduce the temperature of the contents of the piston. So, holding pressure constant, decreasing the volume you also have to decrease the temperature. And the inverse is also true: if you want to hold the pressure constant while increasing the volume, you need to put a bunch of heat into the system.
In summation: it's not that an increase in volume *leads* to an increase in temp. But if you want to keep the pressure constant, each increase in volume needs to be accompanied by an increase in temp.
