During the presidential election, candidates go from state to state speaking to the people in order to convince the people of that state to vote for them. A "battleground state" is one that can't be predictably democratic/liberal or republican/conservative. These are the votes candidates fight for the most because the people are more easily swayed one way or another. The state that I live in is predictably democratic/liberal so it's not a battleground state. Our electoral votes ALWAYS go to democratic candidates. Not all states can be predicted based on history. In this case, "criss crossing" means traveling back and forth. Okay? I couldn't think of an easy way to explain it. Maybe someone else can explain it in simpler terms. sorry.

criss cross is a verb and probably means in this sentence something like going back and forth in some place (in this case, the (US, I guess) states) battleground states is metaphorical...the states are like battlegrounds or battlefields for something...they are the place where a war is going on...