I have read interesting facts that say skilled chess players think multiple moves ahead, with some grandmasters even thinking over 10-20 moves ahead, which sounds insane.
I am a bit confused as to what thinking moves ahead mean? Like, do you think you own future moves ahead, or your opponents or both?
I would guess that in order to think what future moves ahead you will make, it would depend on the opponents moves? In that case, how is that even possible, since theres literally hundreds, even thousands of moves/combinations that can be made over the next few rounds.
Or does the person make an assumption on what move the opponent will make, and then think of a move in response and so on
we think of the moves our opponents are most likely to play, and how we reply to it. normally you should see at least 3-4 moves ahead in a 300 rated game
300 rated players see 3-4 moves ahead ? 
ya kiddin right ?
Calculation is one tool... in other words it's not the only tool.
In positions where it's useful (and possible) of course GMs will look 10 or 20 or etc moves ahead. If you can play a blindfold game then you can "look" a whole game ahead, and players far below GM do that.
The strength of a player's ability to calculate isn't depth, but efficiency and accuracy. I might spend 20 minutes trying to figure out a move that a GM would dismiss as incorrect right away... and this implies that even when I'm calculating the correct line for the correct reasons, I'll be distracted by bad moves that a GM wont waste time on.
And this is the kernel of truth in Capablanca's quote "I only see 1 move ahead, but it is the best move." In many positions all you need is the correct evaluation and the correct idea.