19.a5: "The white rook now has more space." So what? He can't do anything with it. Anyway, it's a good lesson in simplification for beginners.
Victory by Attrition
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"Attrition warfare is a military strategy in which a belligerent side attempts to win a war by wearing down its enemy to the point of collapse through continuous losses in personnel and matériel." - Wikipedia
The same applies to chess: when up in material, it is best to evenly exchange away all remaining pieces - obviously, this means that when your opponent only has a king left, you have something else remaining with which to checkmate them. If your advantage is a single minor piece, this obviously isn't enough, so you have to take care to have at least one pawn as well.
The following game, which I recently completed, is a good example of attrition. My opponent made a single blunder and I capitalized with a surprisingly easy chain of attrition exchanges: