IMHO, I think that the weight of the countertop may put stress on these spots of the cabinets and cause those areas to break off.
If your cabinet material is particle board, the glued fibers of particle board tend to "spread" when screwed down into. (are these IKEA cabinets?)
I'm unsure from the photos if it was a case of too big screws or that your contractor missed his mark? It appears that instead of screwing into the center of the upright wall, he was too close to the edge, or the screw went in crooked.
Either way, I would bring him back to rectify this. It may require another face plate attached to prevent that "bulge" from breaking off and compromising your countertop.

Didn't predrill, and /or off center, likely just used drywall screws. At the very least remove those screws and redo properly a distance away. Ideally the top should be removed, those areas glued and clamped, then put the top back on predrill and use particleboard screws.

Is it just sloppy work? I'm not feeling very confident about his skill level.

Being paranoid as I am re: sloppy, good enough for now work, if this fellow installed your uppers, I'd be looking at them just in case.
Now, I realize this is a cabinet top installation, but in uppers there's a strong area to screw to the wall, then there's, well, not strong. If drywall screws were used, that's really not strong. They sheer.
I'd just check...

What Jakuvall said! Speaking from my (in)experience, that looks like the screw was too close to the edge, and probably not pre drilled. Not that I've ever done that myself, of course...

My best guess is once the strain caused by the screw is released (I.e. the bulge happens), I don't -think- it would spread since there is no further strain acting on the cabinet. But I'd still want them fixed. It's not like he just missed one or two, six is a bit much.

Could he use small L brackets on the underside instead since he's having an issue? I can understand needing to secure the top, but the weight of the counter should pin it in place pretty good. I bet it doesn't need a ton of screws, just enough to keep it in place for install. I could be wrong, but someone here will know for sure.