It looks bad for White - how can he save his e6 pawn? It has no support from White's pieces and can be taken in two moves - ...Kd6 and ...Kxe6. Pillsbury had seen this too, and had a plan to bring up support.
White has two connected passed pawns - a powerful force. But they are stopped by Black's king. Meanwhile, Black is making a passed pawn on the queenside and can also make one on the kingside (by playing ...h5 or ...f5). It is a law of physics that the king cannot be in two places at once! So the game turns into a race.
Let's take a break and look at this strange position:
Normally a game of chess is a combat, with players fighting against each other. But this one is different - it has become not a battle, but a race. Both sides are trying to get their pawns to the other side of the board first, and have pretty much given up on trying to stop the opponent.
White has five pawns against three. But this is not so important, since three of his pawns are stopped by the black king (which is why 36...Ke7 was the best defense). Black meanwhile has a pawn on a3 which the White king must watch, and another passed pawn will be created on the kingside, by ...g4, ...h4, and ...g3. White cannot stop that one, so it will be a race with his one free passed pawn, the b3 pawn.
Because of this, Pillsbury won clear first place in the greatest tournament that had ever been held. He became known as one of the best players in the world.