The biggest reasons to care about governors in other states are threefold to me:
1. Redistricting - States do redistricting. In most states the governor has a big influence on that. Even if the legislature is of the other party the governor can veto. There are states that now have commissions to do this but in general redistricting is at the state level and is hugely important as it makes a big difference in who controls Congress which affects us all.
2. Voting Rights - See above. Governor usually has a big influence on what the legislature does with regard to voting. Not just who gets to vote, but how and when. Think about things like voting hours, locations, etc. Again makes a difference in who controls Congress.
3. What happens in one state can affect the rest of us. If State A is badly run and needs a lot of federal help/money that can affect the rest of us in many ways. Also people may want to move out the state. And, if I was thinking of moving it sure might make a difference in where I chose to move.