How does temperature affect volume?

In general an increase in temperature will produce an increase in volume (apart for few materials, Polymers-like I think, that gets smaller and are used to seal electrical cables).
The increase in temperature means an increase in Internal Energy that, in turn, means that the atoms of your material vibrates more thus displacing more from their equilibrium position and so needing more space/volume to vibrate. The overall volume of the object will be bigger.

Imagine, as an example, a platoon of soldiers in formation; the platoon is stationary but the soldiers move a bit. Now imagine that the soldiers start to shake a lot! They will occupy more space.

For a constant pressure, we can use Charles's Law, which states that,

#VpropT# or #V_1/T_1=V_2/T_2#

So, that means that volume is directly proportional to temperature. Even then, since we increase the temperature inside a material, the molecules' kinetic energy increases and they start to vibrate more and move around further from each other, therefore accounting for an increase in volume.