Yep this is the method! So:
1: ctrl-shift-click the brush selector and choose the lasso select instead of the rectangle select.
2: go to your canvas, control-shift drag to start drawing a shape around those spikes. The shape will be green. Begin holding down alt as well, and it will turn red, meaning it will hide whatever you draw around
3: delete hidden as the guys said.
Another easy way is to mask the stuff you dont want, go to the split section of the subtool palette, and split masked. That will break off the unwanted buts into a different subtool. Then you can delete, hide or merge that other subtool.