I'm going to Hannover next year for a year abroad. You apply in your first year and can choose a first and second choice as to where you'd like to be placed. My course will be taught in German and so I need to be able to speak and write it to pass (you only have to pass for Durham's purposes, otherwise your degree simply goes back to straight Law instead of Law with European Legal Studies).
Some of the universities in the scheme do teach in English (Groningen definitely does) and I think one of the other ones does.
In terms of cost, you don't pay any tuition fees to either Durham or the foreign university for the year. You are still eligible for a student loan and I believe you get a grant from the European Commission of £4000, although I could be wrong. I'm hoping I'm not. Also, when considering living costs, note that these will vary significantly between the different places: Germany is very cheap whereas Switzerland is very expensive.
You apply for a place in your first year and will have an interview (very informal) about why you want a year abroad and why you want that specific place. My friend didn't stress enough that she wanted to go to Germany (one of the more popular choices) and so she was persuaded to transfer to Groningen, which she didn't mind in the end.