The king is at C7 and the Queen is at A8 in the video. They king can't do anything. The whole reason to under promote is for white to avoid stalemate when he/she has a winning checkmate.

The only other moves are after white pawn b7 would be to just run the king out of the fork (legal) but then black's whole position collapses and white just cleans up the same as if black plays king B8.

Once the king is tucked in and the D file pawns are blocked he can ONLY move to C7. If he moves out of the fork earlier to C7 or D8 he still loses the rook and the two pawns are still trapped. Against white's D pawn.

edit: We're also talking about this puzzle instead of the original. https://www.youtube.com/watch?v=iSKtG-8TwI0 Which is different and doesn't use old rule loop holes.