Went to https://asciinema.org/ and clicked the demo video. I'm a bit confused because it looks like future lines are being rendered after the cursor while lines are being typed. Is that...correct? It looks like either a blatant bug in their product (that they're displaying on their homescreen demo!) or on purpose for some reason I can't find.
Ohhh this makes a lot of sense to me now too, I also thought it was an mp4 until reading your comment and thought you were totally insane to have "watched the video" wrong lol. I see now it's an animated terminal rendering via HTML elements, very cool