To use the grapple claw, hold up and forward while jumping at a wall. So if trying to stick to a wall on the right, you would hold up and right. That will make you stick to the wall until you press jump again or stop holding up.

For some reason this took me a while to figure out, so I'm putting this here for any others having trouble.

Any idea how that might work with an Xbox 360 controller? I'm not that far into the game yet, but it might be good to know in advance.

360 pads work the same way. Hold the left stick or directional pad up and toward the wall to cling to it.

I figured out how to stick to the wall with the grapple, but what's the point? I actually find it to be a detriment actually. You can't climb up the walls because when you jump you angle downward and go flying far away from the wall. Before I had this I could make some bouncing jumps off a wall and reach a higher platform opposite, but with this, you can't even make those kinds of jumps. I've yet to find any use. Wish I could disable it or something.

there are a few places where you'll need it, but overall yeah it's a pretty useless item.

Well pressing up doesn't actually affect the jump in the slightest anyway, only left/right and the duration the jump key is pressed. If you're having trouble making jumps around corners without it, just avoid hitting up unless you actually need the claw. Once you get the feather the claw becomes less important, but there are still quite a few areas where it's useful. In some boss fights it can be very excellent for avoiding projectiles, for instance.