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.
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.
Are...Your fingers boneless? For awhile I kept my fingernails cut probably far too short and didn't have too much trouble pulling the occasional arm hairs. That said, point stands regarding picking up smaller stuff, and I appreciate the amusing visual and nail enthusiasm!
Every other mammal has finger/toenails, most of them make very good use of it, others not that much (e.g. elephant toenails look pretty useless). Organisms don't necessarily keep inherited traits because they're useful, but because they're not very detrimental.
I recall reading something about how they give you more tactile sensation on the part opposite and let you expert more pressure but it was a little while ago, might be hearsay.
Because fingernails (or early versions of it) helped our cladistic ancestors survive hundreds of millions of years ago, maybe more. Maybe it helped them find food or protect themselves, or some other reason that isn't so intuitive. Certainly they helped us (as humans) to survive, but the reason we have fingernails right now is just a happy accident.
Many people in this thread or conflating what we use them for today, right now, as reasons why we have them, which isn't exactly right. Certainly they are multifunctional, especially in a modern context, but that is not the reason why we have them. All these reasons posted in this thread remind me of Psychological Evolutionists that only use logic and intuition to find reasons to why we do things the way we do, but in reality it often just happens to be that way. There is no reason for evolution other then happenstance and happy accidents.
tl;dr - We didn't evolve fingernails to pick up dropped coins, our cladistic ancestors evolved them (likely many hundreds of millions of years ago) for some reason that I don't know. Maybe to dig things up or defence. Who knows. Any way we use them today is likely just a bonus.
P.S. I don't like using evolved as a verb, as if it was a conscious undertaking. It arguably isn't, not really.
I remember reading of a special treatment that makes them sturdy like claws. I won't mind that. A multitool made out of your fingernails can be pretty handy.
Day 16 with claws. Wrecked the bedsheets again tonight. My nose is constantly bleeding because I keep picking it in moments of relaxed forgetfulness. My butthole burns like I ate chilis because I still haven't figured out how to wipe it - my family has learned to live with the smell while I keep practicing yoga to eventually be able to give it the cat treatment. Gwen still refuses to be touched after the incident last week... But even the desire to be caressed won't convince me otherwise. This is me. I am a weapon to protect what I love. My body, my choice - and threats of divorce won't get between me and my constitutional rights to protect my family.
I encountered this in some paper encyclopedia as a fun fact and I struggle to google it now, so probably it's BS. I recall people who make world records out of their nail lenght use acryl and oil, but it doesn't seem to help that much. The base of the nail seems to be the weakest part of that scheme, and even a metal claw would probably get torn from the finger under any serious pressure causing more problems that it solves.
They allow us to more precisely manipulate and feel with our fingertips by providing a hard surface to work against for the otherwise fleshy fingertips, and can themselves be used to pick at things smaller than fleshy nubs would be able to. That's not to say fleshy things can't be super precise and manipulable, but evolution doesn't work that way and we already have these grasping hands and not tiny worm fingers or something
Same reasons monkeys have them; it helps with grabbing on to branches. The stiffness of the nail results in an increased grip strength, because the forces are blocked.