What does autopilot mean in chess? I've seen people saying things like "from now on this is just autopilot" or "autopilot from move 20"
I think overtheboard is mostly right, especially because your quotes seem to have this gist, but there is another aspect to playing on autopilot in chess, more often referred to as "playing by hand", i.e. without using you head. 

If you reach a certain level in chess, in almost all positions a "typical" move represents itself to you, not necessarily the best move, but with high probability a decent move. This phenomenon is closely connected to the term priyome, which refers to standard positional responses and methods, that somebody with a certain amount of chess culture just knows. 

So if somebody talks about playing on autopilot or playing by hand, he might just mean choosing natural strong moves without looking deeply into the position. Often this is enough to beat weaker players and it can be done in all kinds of positions, not just in easily winning ones or in forcing lines.