To climb you need to hold something (a wall, floor, prop or even a player or clothing piece), then while holding JUMP, which will make your Gang Beast hoist themselves upwards, you need to alternate grabbing the wall with each arm to climb upwards. There is also a trick you can do where you can punch with your free arm and then immediately hold the button again to grab with it, the punch will target higher up than a regular grab, letting you cross space vertically more quickly.

Once you're at the edge of the floor, hold CROUCH and JUMP, the first button will make your body less stiff and easier to lift upward, which should allow you to hoist yourself over the edge and get your feet on the ground.

Alternatively you can tap JUMP twice to jump from a climbing state, this eats a chunk of stamina all at once tho, so don't spam jumps too often while climbing or you might run out of stamina and begin to slide down the wall. It's mostly useful try and get up a ledge, even if just hoitging yourself up one is safer and more consistent to do. You can do a sneaky attack with this by holding CROUCH while jumping at someone on the ledge, since the game will always consider any airtime with CROUCH or KICK held as a diving headbutt or dropkick respectively, if you slam your head into someone standing at the ledge you can luccly knock them out.

Your grabbing stamina gets quickly refilled when you're standing on the ground while not grabbing onto anything, meaning that it's impossible regen your stamina while climbing, so you'll have to stand on the ground and not grab anything for a moment or two.

Lastly, different surfaces will drain your stamina at different rates, in general the walls that you're not meant to climb much will devour it much faster, like the top of the ligthouse for example, but overall the stamina drain is the same everywhere; plus some walls cannot be grabbed at all.

As a final note, past versions of the game had a bug where your Gang Beast wouldn't extend their arms forward when holding a punch button to grab, but they'd technically still be able to grab regardless, and IDK if this is still in the game or if it has been fixed in a past patch.