I really struggled with this one. I saw that in the 1... d3 line, Black's king was just in time to protect his d3 pawn from white's king after exchanges. But what I didn't understand is what would happen next. But now I do: White's king is stuck on d2, holding back our passed pawn. So the the queenside and kingside pawns will now have to march forward. This will eventually lead White into zugzwang; his king will be forced back.
The Process of Decision Making in Chess Volume 2: Practice positions and solutions. Position 8.9
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maybe you could do d3 and then when the pawn takes it, you take it. Then, when the king is getting close to the pawn, then you exchange pawns on the a and b columns. If you do that right, then you will have on pawn that is defending the original pawn. Then, the king is unable to get it and so then you move your king up so that it is near the king and also so that you can defend your original pawn with your king. Then, the king will be forced to move out of the position blocking your pawn. Then, you can get a queen.