PID controllers are one way to control systems described by differential equations.

Given a certain desired value of the solution they can be used to force the dynamics to converge towards that value.

For certain types mostly linear equations they are the optimal way to achieve this.

I have no idea why, but the topic is almost never presented that way. Instead people talk about valves and pistons and whatsoever, but in the end it’s just that:

You have an ODE and some desired value for the solution ; you have some control over the inhomogeneous part. PID control is just a tried and well understood way of excepting this control.