What is a partisan stance?

"Partisan", in the context of politics, means "of or for a party". That's it. So, a "partisan stance" would mean approving of the position of a political party.

In today's environment, "partisan" often is used to mean "unreasonable" or "uncompromising". Thus, to accuse someone of taking a partisan stance means that they back a certain political party's policy, and refuse to consider a compromise or alternative. It carries the negative connotation that the person is being unreasonable or obstructionist.