These instructions will be filled out with more detail.
1 - The tile
This is the Spectre monotile.
Or rather, it's one possible Spectre monotile. The important facts are that it has 14 vertices with a particular pattern of angles, and the curves joining the vertices don't have any reflection symmetry.
2 - The Mystic
The first step in building up the tiling is to join two tiles together, like this.
The authors call this the "Mystic", because it looks like someone sitting in the lotus position.
3 - The metatile
The plan now is to join tiles together into bigger units called metatiles.
Nine tiles join together to make the first metatile.
4 - And so on
The metatiles join together to make an even bigger metatile.
You can continue this process forever, so the Spectre can tile the plane beach!
Working out what goes where can be difficult. The authors noticed that you can draw the same quadrilateral on each metatile, to help you get your bearings.
(IMAGE TO DO)
You try!
I've made an interactive simulation of the tiling process.
Try fitting the pieces together. Don't worry if you make a mistake - it's just sand, so you can rub it out and try again!