Math! Also, noise!

There are algorithms (a set of math steps) that make pseudo-random numbers. These usually involve large prime numbers, because those usually generate fewer repeating patterns.

A truly random number generator is similar to rolling dice: you use some source of randomness and convert it to a number. All electric circuits produce "noise" (which is often received radio waves and such that interfere with the circuits). Think of tuning a radio to a channel with nothing on it--you get "white noise", which can be a good source of random information. Then all you need to do is convert that to a range of numbers, and you're good to go.

These are fairly simplified explanations, so take them with a grain of salt, but they give the general idea.