I think the OP is asking why anyone would care who the governor of another state is. Seems pretty clear that the governor of your own state is important.
A few thoughts come to my mind:
- Travel. If you're ever going to travel to that state, you are subject to their laws. Maybe they discriminate against your kind, or don't enforce laws well which encourages more crime, making it unsafe for you. Crime can also bleed into surrounding states.
- Environment. Air and water doesn't just stay in one state, if pollution is unchecked it affects everyone, especially those downwind and downwater from the state.
- Economy, corruption, etc. If a state is mismanaged, will the federal government have to bail them out? Do they give unsustainable tax breaks that take away industry from other states?
- Immigration enforcement. Not going to get deep into this but if they are a border state, what they do clearly impacts a lot of people.
Is that enough?