What was Big Willy Shakes going for?
Grumio loves the attention that Curtis gives him. In The Taming of the Shrew, language is often associated with power. So, if Grumio has the deets on what went down between Petruchio and Kate on the way home, he's got the upper hand. He makes sure Curtis knows it, too, by telling him he's too tired to tell the story. But then he goes and ruins it by telling him anyway. Oh well. Sometimes telling a tale is just too tempting.
When he says "thereby hangs a tale," he's literally talking about the story that Curtis wants him to tell. It's mysterious and dramatic. No wonder Curtis wants to know what happened with his master. Grumio builds it up and makes it sound really important and exciting.
People in Shakespeare's day said "thereby lies a tale" so Grumio is playing off of that old saying. By changing it to "hangs," he puns tale and tail. He's saying both a story and a backside. You see, the story he's telling has Kate falling off a horse into the dirt. Grumio says that's where the horse's tail hangs and there's a good story about it.
After Shakespeare used it here, he put it in other plays as well, including Othello, The Merry Wives of Windsor, and As You Like It.