A leash is a rope that you hold to keep your dog from running off when you go to a park. The phrase can also mean "To keep under control". So if someone says to you "Put that dog on a leash!" they are saying to put a rope on your dog so he is under your control. A leash can be used for other animals and even small kids for when you go skiing. But if someone says "Those kids need to be put on a leash" they are saying "Those kids need discipline/The parents need to make them behave"

Leashes are for dogs, so if you keep someone (something) on a leash, you mean that person is kept under control... as if he/she were a dog on a leash.