Cut your fingernails way too short, then try to pick up something small. Not even with the nail part.
The hard flat back gives your flesh something to press up against when pinching. Imagine pulling a hair out of your arm with your fingers. Easy, right? Now imagine pulling a hair out of your arm with your lips.
As to why they grow and wear out and break, what the hell else should they be made of? Hands take a beating and you're better off with a material that grows back when it's worn away.
And finally, why do we have toenails? Mostly because they used to be leg hands. Your toenails actually grow about 3 times slower than fingernails, so that might be evidence of how much less useful they are to us.