It's not easy to explain PSO using mathematics (see Wikipedia article for example). But you can think like this: the equation has 3 parts:
particle speed = inertia + local memory + global memory
So you control the 'importance' of these components by varying the coefficientes in each part. There's no analytical way to see this, unless you make the stocastic part constant and ignore things like particle-particle interation.
Exploit: take advantage of the best know solutions (local and global).
Explore: search in new directions, but don't ignore the best know solutions.
In a nutshell, you can control how much importance to give for the particle current speed (inertia), the particle memory of the best know solution, and the particle memory of the swarm best know solution.
I hope it can help you! Br's