"Why does one of the parties fight against people voting?"
In general, high voter turn out leans left.
And each state determines the rules for voting. I'm in Illinois, so when I register I'm assigned a polling place. When I go there to vote, I give them my name (no need for ID) they find it in a book, I sign a thing and they give me a ballot and tear off a receipt that I keep. I'm in a traditionally Democratic/blue State.
Republican/Red states tend to do what they can to make voting harder, like frequent purging of voter registration (and not notifying people that they're not registered anymore) or making it nearly impossible to vote by mail, then require voting in person and limiting the number of places you can vote to the point people stand in line for hours. Oh yeah, then make it illegal to offer people waiting in line a bottle of water.
Huh?