That is a great shot, Nick!
There are two versions of this cassette case - the earlier version, as used on the show, has 3 small hinges, and no rubber feet on the bottom (same type of hinge and mounting as on the Game Boy cases). The hinges are attached to the case only by thin plastic flanges and seem not very strong - this version of the case cannot open fully flat and forcing it could cause the plastic around the hinges to give way.
The hinges themselves are only held into the slots by a single tooth stamped into the leaf which bites into the plastic. To me those screws look like an attempt to reinforce the hinges and keep them in place after either the leaf's tooth no longer worked, or the fragile plastic slots broke away. I'd always wondered how they kept them from breaking and this seems to be the answer.
It's possible the screws might also serve to retain the acrylic inserts, but it is tricky to drill into the edge of 1/8" acrylic, and they would have to be rather tiny... are you thinking they act more as a clamp, @RustyH? I'd always assumed the inserts were just glued into the case.
A later (and much more common) revision of the cassette case saw those three hinges replaced by a single long piano hinge. The mold was adjusted so that the case would open close to flat with less resistance and rubber feet were added to the "bottom" of the case.