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.
There have been studies conducted on this (by de Groot and later Herbert & Simon) where they showed both masters and novices chess board layouts for a few seconds and then asked them to reconstruct the layouts from memory. Some of the layouts were real chess positions and some of them had pieces randomly placed, sometimes in illegal positions. The masters were vastly better than the novices at recalling the real positions, but virtually no better at recalling the random layouts. This demonstrated that masters perceive a position based on the familiar chess elements of it - pawn structures, tension between pieces which can capture each other and so on.
Here's a video of GM Patrick Wolff being given the test and talking afterwards about how he was able to recall the real positions.
Most people imagine the position in their mind. When you get better at chess, you get better at imagining it in your head. Remembering all the moves is technically possible, but imagining it is usually the easier option if you're not in an opening you've studied relentlessly and you're 100% certain. I know it can feel like you'll never be able to develop this skill, but I thought the same way as well for a while and now I can go up to 40 moves while keeping the exact same quality of play as normal. Good luck!