If I wanted to make similar pieces to this, how would I do it?
How does the artist get the background colours like that?
It's just simple linework on one layer, painted over on another (though it looks like they did it on paper, so no layers, but the principle is the same). The background is just a simple gradient.
There's nothing there that's particularly advanced.
It looks traditional to me, possibly colored ink with pen linework. If so so, contrary to what >>2369532 said, its pretty advanced because getting a smooth gradient like that in the back ground is very difficult.
This is a piece by the late Moebius, in case you missed the signature. In this piece he used a pen for the inks and watercolor for the colors. If you wish to replicate his style with digital media keep in mind there is a lot of subtle texture in the colors which you will miss out on if you use standard soft airbrushes and gradient fills.