How does this work, are the people able to really solve the game this fast or are they memorizing sequences? Is it possible to see the game before the click starts?
Memorising sequences is not the right way to look at it - it's simply a result of practice (or playing it too much): More like muscle memory where you see and act without consciously thinking about it. Watch speed solving Rubik's cubes to see how they do it on an instinctive level.
With Freecell, once you know what you're doing you know at a glance what your options are: how deep a stack you can move and which cards are key to clearing out and in which order.
As well as playing way too much Freecell back in the 90s I played Minesweeper to death. Probably a better example of how you can progress from needing to consciously work out where you could safely click, to knowing at a glance, to being able to click the one cell that would open up the area. If you were actually flagging bombs you were going too slow.