(Note that here we are talking about exchanges that are like-for-like, i.e. minor piece for minor piece, queen for queen, etc. Captures that lead to material imbalances are another, more complicated subject. If you are interested in that, I recommend this article by GM Larry Kaufman on NM Dan Heisman's site.)
For a general overview of the topic by a strong player, the Chess.com video General Strategy: When to Exchange Pieces by GM Dejan Bojkov is well worth the time, as it goes into various strategic guidelines for exchanging in a clear fashion.
Below I've provided a few game examples in order to highlight some of the situations where piece exchanges can make a significant difference in the course of the game. There are a considerable number of different lessons to be had, but as a fundamental concept, I believe that if the improving player asks the questions "how will this exchange affect the current position?" and "what are the strategic implications of the exchange for the rest of the game?" before exchanging, that in itself can help provide a measurable boost to chess understanding and playing strength.
1) Samisch - Nimzovitch (Berlin, 1928)
This classic game was included by Nimzovitch as game 13 in his book Chess Praxis; a more modern commentary by GM Raymond Keene can be found at the link above. Here Nimzovitch deliberately exchanges off both of his bishops in advantageous ways that are central to his strategy.
2) Hou - Kosintseva (Khanty-Mansiysk, 2014) - original full commentary
3) Ramirez - Shankland (St. Louis, 2013) - original full commentary
4) ChessAdmin - Expert (Annotated Game #84, "Piece exchanges and draw offers will lose you the game")
5. GM Yermolinsky - ChessAdmin (Annotated Game #4, "GM Alex Yermolinsky simul")