How would constantly growing nails have aided early human?
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 &mdash; 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) &mdash; 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.