In an article I'm reading, it is said that the moduli space of conformal structures on a disk minus $k+1$ boundary points is a ($k-2$)-dimensional manifold. I want to understand why and have done some quick research, and it seems I need to look into notions related to Teichmüller theory; however, I know nothing about these things, and I don't know what to search for exactly. I've glanced at Farb & Margalit, "Primer on Mapping Class Groups", but all the compact surfaces they consider are punctured in their interior, not on the boundary.
I would be really grateful if someone had a proof (if this is an easy, well known fact), or a reference for this.