I knew a genius kid at school could solve one in well under a minute.
I actually was surprised looking that up, I thought the record was like 5 seconds before.
The speed solving is memorization and then execution. Generally you memorize a lot of algorithms (somewhere around 70 or so, I forget) and then based on how certain parts of the cube look (particularly the yellow side) you apply the correct ones (it's more than one per solve). Of course, the other part is execution. Just a single incorrect movement and chances are you've knocked yourself out of sequence.
And then tons and tons and tons of practice!
And a good cube. There are a lot of models of 3x3x3 cubes (aka, the basic "Rubik's Cube") that are specifically built for this speed solving. The actual official Rubik's Cube would probably explode if you tried those techniques on them. Particularly those models that were around at the time when the thing became a brief fad decades ago. I still have one of those somewhere, it makes horrible squeaking sounds.
And no, I cant do that speed stuff myself. My method only uses 3 algorithms, so most of it isnt memorization. It takes like 10 minutes, sometimes longer for me to do it. Nobody would use that screwy method for speed solving.
The part that surprised me with all of this is that it isnt nearly as hard as it looks (the normal solving, not the speed solving). I used to think that the basic cube was like, this unsolvable mess. Like, how could anyone possibly figure that thing out?
Now it's no challenge at all. Well, the basic cube anyway. The other ones escalate in difficulty depending on what they do.
I could talk about these all day.