Actually, I believe that the person speaking is probably talking about some machine or device that they have which is leaking.
A person whose car is leaking oil might very well say "I think I'm leaking oil." Obviously, the person isn't leaking oil - it's the car.
Likewise, if some battery or other device that they own is leaking out the edges or seams, I might expect to hear "I'm leaking out of my edges" = "My machine is leaking out of its edges."
As both Eliot and Jesse said, context would certainly help.