The monster is freaky-looking all over, but some of his freakiest-looking bits are his heavily lidded eyes.
Most of the monster's appearance was designed by make-up artist Jack Pierce (see Method of Production). But the heavy lids were provided by none other than the guy portraying the monster, Boris Karloff himself. Karloff says:
We found the eyes were too bright, seemed too understanding, when dumb bewilderment was so essential. So I waxed my eyes to make them heavy, half-seeing. (Source)
The heavy eyes, then, are supposed to symbolize, or show, that the monster is bewildered, and doesn't understand what's going on. That's a big part of what makes the monster sympathetic; he's not responsible for what he does.
Who knew wax on the eyes could say so much? (For that matter, who ever thought of putting wax on their eyelids? Ick. Way to sacrifice for the film, Boris.)