I read the following: "The Klein bottle contains a copy of the Möbius band". I assume this means that there is a subspace of the Klein bottle that is homeomorphic to the Möbius band. How do we obtain the Möbius band from the fundamental polygon of the Klein bottle?
Here's a representation of a Mobius band. Hmmm, what does this suggest...
If you don't join the red edges together you get what you want, so you can cut the Klein Bottle along the red edge (or what it becomes in the Klein Bottle when the two red edges are identified).