It means waiting for something bad to happen which you are expecting. It comes from a famous music hall joke about a man who is woken by the drunk upstairs dropping his shoe. He can't get back to sleep because he is waiting for the second crash on the ceiling. Eventually he shouts upstairs "For Heavens sake, drop the other shoe!"
The basic expression came from apartment living. A person living in an apartment underneath another apartment would hear a person take off a shoe and drop it on the floor and then wait for the other shoe to drop. Then the thump of the other shoe would occur. Then the person in the bottom apartment could continue reading or whatever without interruption. However, it became a useful expression if two irritating events frequently occur together. If someone says, "Why didn't you get back to work?" Rather than say, "Any time that guy comes down the hall pushing that noisy cart that goes clang, clang, clang, with that horrible sound, and disturbs everyone, a second one comes right afterward with its own irritating racket," you can say, "I'm waiting for the other shoe to drop."
