A proof game is a legal (though possibly weird) chess game
reaching a given position, thereby proving that the position is legal.
One often solves a retro-problem by finding a specific proof game
for the given position: this can be used to prove that e.g. castling
is allowed, or that White may have the move, that an en passant capture
needs not be allowed, or that a draw by the 50 move rules did not occur.
Generally, when one does not see why a given position is problematic,
it is often a sensible idea to try to build a proof game. Difficulties
It is important to understand that, regarding many issues, proof games
can only prove one side of the issue. E.g. proof games cannot be used
to prove that castling is forbidden, that White can't have the move,
that an en passant capture is legit, or that a draw did occur. For these,
other lines of reasoning are required.
Shortest Proof Games
are a very different topic...