Hmm.... there isn't one right way.
In fact, at the other thread "catapult" (if you are able to sift through all the arguments and unhappiness), you would notice that there are different types of catapult.
eg. the one that is used by my teams, and there are people that also listed crossbow/slingshot as catapult... and also the trebuchet-style type of catapult, etc.
So you got to see which method suits your requirements, and then you find out the details on how to build the type of catapult that you want.
Anyway, if you are looking at 8059a, they are using slip-gear method - basically the motor will turn one way to coil up the rubber bands, and then the ratchet will hold it in place.
And to release, it just turn a bit more to the part of the gear this is shaved off. It works similarly to how slip-gear works in linear puncher.
You can try to google or youtube about how slip-gear works. there are lots of people talking about it during nbn season.