This pid-stuff is needed to get a good compromise between settling time (which is the time needed to accept the new control setting), ripple (which is the oscillation arround the correct control setting) and precision (which is the deviation from correct control setting).

Why is all this needed?
Many systems to be controlled show a relevant inertia. Means, they need very long to time to accept a change of control setting. In many applications it cannot be waited so long. So, a trick is used to enhance settling time: A modified 'control setting' is applied, which is higher than needed.

Let's have an example:
Assume you have a really big heater showing a heavy inertia. Running at 100W means a control setting of 1V. Now you want to change the heating power to 200W, according to a control setting of 2V. If you now apply 2V, then the heater needs long time to settle to 200W. But if you apply for a short period a higher voltage than 2V, let me say 10V, as if you would want to change the heating power to 1000W, then the time needed to reach 200W will be considerably shorter!
Of course, you must time this oversetting of 10V very carefully, otherwise you will have a big overshot of heating power, resulting in much more than this wanted 200W.

A PID-controller now has three modules reacting to three different characteristics of input signal. There's a 'proportional' section, an 'integral' section and a 'differential' section. In our example it's the differential section which gives you the brief overshot. For this it 'looks' for fast changes and produces a control signal, which is proportional to the step change.
And to get this information about change rate, etc., just filters are used. As they contain resistors and capacitors, they offer a comfortable way to implement the differentiation and integration. Remember a RC-low pass filter provides integration and a RC-high pass filter provides differentiation.

It can be very sophisticated to get the best PID-coefficents for an application. For this, it must be clearly known, what the actual requirements are for settling time, ripple and precision. Normally, getting fast settling time means an increase of ripple and a decrease of precision. On the other hand, getting low ripple and high precision means long settling times.

Kai