Thanks for visiting! I agree with the difficulty of finding the mechanics, it was pointed out by others as well that we should have explained them in more details (they are roughly outlined in the Objectives panel).
In case you are still interested in another try: you can move on the ground by using A and D or the arrow keys, and as a spider, you can walk up walls (it's a bit sensitive mechanic, so might take a few tries to get it). From the second level on you can also create web by clicking on the area where you would like to attach your web, and then you can swing with WASD or the arrow keys to reach the next platform. :)