He says he "calculates it better without looking at the board"

To what extent do you think this kind of skill is natural and how much can it be learnt? And is it learnt simply by playing or can you do mental exercises that help? I have a terrible short-term memory, but I think it's because I don't absorb information properly in the first place. For instance, I have trouble remember people's names because I don't take it in why it's mentioned. I never forget a face, though, so I suppose I do have a good visual memory.

Do people who can do this see the board in their mind's eye or do you remember the moves leading up to the position?

From the accounts I've read, some masters see the board very concretely where their attention is while others have a more abstract image of the board and pieces.

Yes, some players recount the moves at times to refresh and reassure themselves what the board position is.

What practice is good for blindfold play?

Just have a go at it and over time you'll refine your personal process. Try to play a few opening moves. Mate a friend with k+q v. k. If you enjoy blindfold you'll pick up the skills you need as you go along.

Have fun! Mystify your friends!

Morozevich and Ivanchuk are fond of staring somewhere else, and then making a move on the board. In general, of course, it's possible, but there probably isn't a single person who plays blindfoldedly better than "normally".

I remember reading something like that about Kolty too, maybe in Golombek's Encyclopedia, that his combinations were better blindfold.

"Calculating in your head" is something we all do. Even with pieces on the board, you have to be able to visualize things.