Question

What does the using the damper pedal do and (roughly) how does it do it?

Answer 1

A pedal on a piano that when depressed with the foot raises the dampers and permits the strings to vibrate and sustain the tone.

The sustain pedal (the rightmost pedal) is the most frequently used of all pedal, and is also the most essential to playing certain pieces. This is why sustain pedal inputs feature even on beginner level home keyboards.
To understand what it does, it’s worth considering the mechanics of how an acoustic piano works. Essentially, a piano consists of a series of strings, which are hit by hammers when keys are pressed, causing them to ring. When the key is released, the hammer returns to its place, resting on the strings, causing the note to stop ringing.
To prevent strings from other (un-played) notes resonating when notes are hit, a damper bar sits on the strings, keeping those strings deadened.
When the sustain pedal is depressed, this damper bar is lifted. The result is that notes will continue to ring after keys are released- i.e. they sustain.
A side effect of this on an acoustic piano is that the strings of other, un-played notes will also resonate gently, adding richer harmonics to the overall sound. This is known as sympathetic resonance, and many digital pianos now replicate this effect.