When something goes wrong in a circus, they 'send in the clowns' as a fill-in act until they get things sorted out. The song complains about the singer's life going wrong, (presumably in life's circus), so, till it gets sorted out, where are the clowns?
I remember hearing a radio program where Stephen Sondheim (who wrote the song) complained to the interviewer that everyone seemed to be singing it slower and slower, making it sound like a dirge. In actual fact, he said, it should be played and sung as a quick-time number, as was intended when he wrote it.