How do people 'see' the position in blindfolded games? How do most people play blindfolded chess? Is it akin to a live video feed in their mind where they can see all of the pieces, or do they just remember what moves have been made so far and go from there?
You certainly do not remember all the moves made. 

Instead you remember the position on the board. In chess you have certain typical structures, so you don't really need to remember each and every piece's position. Instead you can for instance store the information: fianchettoed bishop to mean the very typical position of pawns on f2,g3,h2 bishop on g2 and knight on f3. Similarly you could store certain typical pawn chains. For instance in the French defence you could have white pawns on b2,c3, d4,e5 and black pawns opposing them on f7, e6, d5, c5, again you'd just store something like "closed French" .

Any player of reasonable strength is doing this automatically and subconsciously when doing calculations at regular chess as well.