From the picture it looks like it's wired for 120V (on the left diagram) and also has the black/red CW rotation swap. I only see the green/white/black going out of the motor box. If there is a red just use a wire nut or tape it off so it doesn't connect with something it shouldn't, it's not going to be used.
Normally in 120V house wiring the black wire is hot, and the white wire is neutral and green is ground.
So, you're going to need some stranded wire and a three prong plug to stick in the wall.......or maybe half an extension cord

, (or an old computer cord or old vacuum cleaner cord or something with a 3 prong plug, 1/2 hp isn't really that big of a draw, most hairdryers draw over twice the power).
In the box with the new switch, just wire the green (coming out of the motor junction box) to green (from your extension cord).
Then wire the white (coming out of the motor junction box) to white (from your extension cord).
And finally wire the black (coming out of the motor junction box) to one of the screws on the switch and the other black (from your extension cord) to the other screw on the switch.
So, green to green, white to white, and the two black ends to the switch. And tape off the red you're not using.
You may have a green screw on the switch, which you can wire into the other greens and the box.