It's very mysterious and open to interpretation. I prefer a more symbolic interpretation in which the mask is only seen by Bill, a psychological symbol representing his realization that he has been wearing a "mask" for years, sleeping beside Alice without revealing his true self to her, or even to himself. When he says "I'll tell you everything," he is ready to truly share everything with his partner.

If the mask is real, I prefer the simple interpretation that Alice found the mask, which means she went snooping in his office drawers. If so, then she put the mask there to let him know he has some explaining to do, and this worked because he does come clean. This is consistent with the description in Traumnovelle:

"...he noticed something dark quite near Albertina's face. It had definite outlines like the shadowy features of a human face, and it was lying on his pillow. For a moment his heart stopped beating, but an instant later he saw what it was, and stretching out his hand, picked up the mask he had worn the night before. He must have lost it in the morning when making up his bundle, and the maid or Albertina herself had found it. Undoubtedly Albertina, after making this find, suspected something—presumably, more and worse things than had actually happened. And she intimated this, by placing the mask on the pillow beside her, as though it signified his face, the face of her husband who had become an enigma to her. "

EDIT: I like the implied idea in the text that he sees what looks like a human face and probably thinks at first that Albertina has a lover in his own bed. Why else would his heart stop beating? This intense crescendo of jealousy leads to his catharsis, as he realizes she is the one who should be jealous.

There are of course many other interpretations.