There really isn't a layman's explanation for a Teichmuller Space, but I can try. I would say my chaps the take on Teichmuller space is imagine you have a complex space. Space-time will work, that is all twisted and bent upon itself. Events could happen at two or more places in time or space. Very bizarre stuff, but could be used as a "hyperspace" notion. Anyways, if you had lots of these twisted space times, the Teichmuller Space would be the collection of all these time spaces that shared the same dimensional properties.
But the idea of many pants sewn together best show what the space would look like in visual layman’s terms. It really involves higher dimensional geometry and complex planes and there is not easy way to explain it but through math.
Comments from Professor Hahn on
Teichmuller Space at Conference 11 on Mern