Ilya Kudriashov makes a lot of those types of maps. My guess would be he uses Illustrator (and or Photoshop ). I'm pretty sure Tim's answer is the one you are most likely to receive.

Now you can of course get away with opensource programs. Inkscape (as an Illustrator variant), and Krita (as a Photoshop variant) would be my choices of the free options today. GIMP may be a little more straightforward to handle than Krita though.

But honestly before you get that far just start making the maps with whatever, even graph paper or plain white, figure out the information you want to include figure out some ideas about how you want to represent it and start from there. Remember those polished final details are the least important step (but perhaps the most time consuming) they are achieved by starting simple making sure all your basics and foundations are covered well and then just refine refine refine.