While various factors have led me to stop following tennis closely, in younger years I followed it quite closely. I remember Bud Collins mentioning the importance of a set's seventh game several times while commentating for NBC during tennis matches, especially Wimbledon. However, I don't remember any explanation of where the idea came from. While researching this answer, I found that BBC commentator Dan Maskell also frequently promoted the idea that the seventh game of a set is of crucial importance, but since I was watching & listening from the U.S., I was unfamiliar with Maskell.
As best as I can find Martin's comment seems to have this correct. The Tennis Abstract blog article about Bill Tilden, states that the seventh game is sometimes called the "Tilden game" because he stated that the game was so important. That article caused me to find a free version of The Art of Lawn Tennis, in which Tilden stated:
In the game score the sixth, seventh, and eighth games are the crux of
every close set.
He then provides an example of a best of three doubles match that turned around when one side had a 1-0 set advantage and was serving at 5-1 but was broken and then fell apart.