My question is how people can hear a performance that differently.
Ignorance... and lack of knowledge. And I don't mean that in an insulting way.
I used to play Beethoven like he did - in performances. I was taught by morons who let me get away with sins on the piano that no credible teacher would let you get away with. I didn't know any better, and my parents didn't do music so they didn't know any better.
Maybe the performance was a one off, however the mistakes don't seem to be a one time problem. He really seems to have trouble with semiquavers and the dexterity of his hand - so unless he had a physical accident that has caused some seizing of the muscles, this seems like he took on a piece that was too hard for him to play. The way he also plays the piano seems inefficient in the way he moves his hands when he changes octaves.
Also, the way he plays the slow movement of the Beethoven seems clunky and isn't soft enough as if he has a hard time voicing the upper voice compared to the triplet accompaniment. Someone with more control over the piano wouldn't have this problem.
Unfortunately my opinion (based on one of his other videos) is that he is playing pieces far beyond his capabilities, as he has another video of the Beethoven Moonlight, which is unfortunately full of mistakes; and this is 5 months prior to the one the OP posted.
Even someone who had good technique and barely knew the pieces could at least play the arpeggios well and much more nimble finger movements. His fingers are too clunky when playing, and he needs to work at refining his technique before he attempts the pieces.
And again, this is coming from someone who used to play like that... and believe me - it's a HARD lesson to be taught; but I'm glad that someone had the guts to tell me so I could begin the process of improving... and because of her I have.