How come it is so difficult in the USA to exert your democratic right?

Because half of the population of the country (republicans) don't want the other half to have any rights at all. They no longer care about freedom or this country. They only care about screwing over those who don't agree with them. Yes, they're that stupid.

Its difficult to vote to keep people from voting. When people are allowed to vote Republicans lose elections.

"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.

Do you have to pay for your ID, and are there any barriers to getting one?

Republicans love to make it hard for poor minorities to have indentification. Hell, I am above the median age but we have to shuffle utilities around between the adults in the home just to renew our fucking driver's licenses here because of bullshit 'proof of address' requirements.

In Europe I get a voting pass sent in the mail for every election. To vote I have to show both this pass and a valid ID.

In the Netherlands it doesn't even have to be a valid ID. If it hasn't been expired for more than 5 years it's fine for voting purposes.

Many European countries (at least the one I live in) have mandatory registration of residences which doubles as voter registration. In my country only homeless people and I think citizens living abroad have to explicitly register to vote.

My understanding is that this is not so in the US, the government doesn't reliably know where people currently live and whether they are entitled to vote, so most people have to register to vote.

In South Africa it's even easier.

There are multiple voter registration weekends 2 times a year. Or you can register any time at an election commission office.

You go with your ID document and register to vote in the district, DONE. You never need to do it again, unless you move to another district.

Come voting day, you pitch up with your ID document and you vote.

As with many American "problems" they just want to solve because it's "too hard"

"Too hard" stopped being a valid excuse in Kindergarten.

In the UK we didn't even need that until recently.

You just turned up, told them your name and address, they tick you off on their form and you vote.

Now you need to show ID as well, which is just not needed. There was no evidence of fraud to start with, and almost all the ID checks are pure theatre. The woman who checked mine at the polling station was about 80. She barely glanced at the passport, but then spent ages squinting at the paper form which was right in front of her. They had a woman who couldn't see checking the IDs. I sincerely hope she wasn't driving home.