As someone who only knows the very basics of chess, what do you mean by "stale board"? Searching for that term didn't turn up any interesting results.

He means that a real GM would know that the position is a draw and would offer a draw and stop playing. The cheater doesn't know so he has the engine keep playing forever.

By a stale board I mean a board state where there are no sensible moves in order to create an advantage. i.e. there are no good pawn moves, no good knight or bishop re-positions, castles mostly stuck and the engine pretty much shows no point difference.

I’m not sure, maybe OP means a board where you would customarily ask for a draw as there is no way for you to win and the opponent could only get a stalemate at best.

I think they mean a closed position without many meaningful moves available. Often times real players make small incremental positional improvements and probe weaknesses, and engines take completely different lines.