Nails grow rapidly and constantly, such that without constant artificial trimming they would reach lengths difficult to manage. How did this benefit early humans, say 200kya? Were they used like claws? Did physical activity keep them at a short length?
I can't say anything for sure (and, with questions like this, it's rare that anyone can), but my guess would be that human nails grow constantly for the same reason as the (analogous) claws of most mammals do: to keep them from accumulating damage.
Your bones, if they suffer mild to moderate damage, are capable of healing through the concerted action of several specialized cell types in your body. This healing process, however, is only possible because the bones are inside the body, surrounded and permeated by living tissue.
Your nails, however, are simple sheets of keratin, and contain no living cells that could heal them, nor any blood vessels that could keep such cells alive. Being exposed to the outside of the body, outside the skin barrier, they cannot really support living tissue that could heal them.
Instead, the way human nails (and animal claws, more generally) "heal" is by growing. Since the nail itself cannot contain cells that would rebuild it, those cells are instead located in the nail matrix, which is located under the nail and from which the exposed part of the nail continually grows out of.
(Indeed, the same is true of most of the exposed parts of your body: your hair and even the surface of your skin itself are also composed of non-living keratin constantly growing from an underlying matrix. This is necessary to maintain the skin barrier between the interior and the exterior of the body.)
The reason human nails must keep growing all the time is that they do continually wear down and suffer damage from various causes (and would surely have worn down much faster yet for our primitive ancestors, who had to sustain themselves through hard manual labor with few or no tools). If your nails just stopped growing after you reached adulthood, they'd be in horrible shape even just a few years down the road, not to mention decades later.
While the rate at which human nails do grow is pretty slow — it takes several months for a damaged fingernail to fully regrow, and up to a year for a toenail (a fact I can confirm from personal experience, having broken my toenail last summer) — it's still enough to ensure that your nails will only bear less than a year's cumulative damage at any moment, rather than every nick and scratch and cut they've suffered during your entire lifetime so far.
In a relatively sheltered modern lifestyle, it's quite likely that even this moderate growth rate may exceed normal wear on your nails, requiring them to be clipped regularly. Still, the fact that you can clip your nails (or, in a pre-modern society, just bite or scrape or cut them down) reduces the selection pressure against excessive nail growth. On the other hand, not having your nails grow back as fast as you wear them down through hard manual work could be quite bad for your fingers, and could easily reduce your working ability or even lead to harmful infections. Thus, it makes sense that evolution has optimized out nails to grow at a rate that, on average, tends to somewhat exceed the typical rate of wear.
It didn't have to benefit them, evolution has no intent and not all traits are advantageous. Not every thing we have in our bodies is there "for a reason", some things are just there, others evolved, and some are vestigial (inherited from our ancestors).
A whale can move his tail up-down only because he's a mammal and can't ever be anything else, while fish move their tails left-right because they're fish and that's their heritage.
Constantly growing nails never helped a single human, but they sure did help our ancestors which had claws and needed them for survival. We lost the claws but didn't lose the constantly growing trait. Basically, next time you trim your nails remember they (in shape of claws) saved your ancestor's life a long time ago and that's why you have them although you don't need them. He needed them.
I can give you my personal account to your question. This can be attested as the truth by many ultra-marathoners amongst us.
Human nails do not grow as fast as your modern society tender hands and feet have made you think, nor do they look so thin and delicate-like in the wild. Human nails that get beaten and worn every day become strong and thick like the talons of a wild beast.
In all my years of ultra-marathon running, I have lost all my nails of both feet many times over. The first time I lost a toe-nail ( yes, it took a year to fully regrow ) it became thicker, then I beat it and lost it again but this time part of the old nail remained glued to the new nail ( more thickness ). The third time the old nail never fell. It became permanently attached to the new nail undeneath. As a result, each nail on my feet now, measures 3mm in thickness. I could never trim them if I liked, because the edges tend to become brittle and fall off like a wild animal's. I could provide a picture if you like but I do not know if it's against the rules of your forum.
Finally, a word of warning. Do not, any of your tenderfoot modern-society nerd who has never run more than 100 miles non-stop in his life dare delete or vote down my answer. This is the most definitive answer to your "naive" question. Let other people benefit from it or at least ask another ultra-marathoner about it.
I find no need to provide references for such a simple question. There are tons of photos on the internet of gorilla and chimpanzee nails and as those are identical to humans' a close study of them would suffice.