The lowest round ever shot in The Players Championship was, until 2017, also the lowest round ever shot in a major championship (which The Players isn't quite, but it's still really important). However, since Branden Grace went ahead and shot the first-ever 62 in a men's major at the 2017 British Open Championship, the records are separate.
The lowest round at TPC Sawgrass in The Players Championship is 9-under 63, a tournament course record first shot by Fred Couples in 1992, then again by Greg Norman in 1994. Both of those players went on to win The Players in those respective weeks, back when the championship was originally played in March.
However, since moving to May in 2007, there have been six rounds of 63 to match. Roberto Castro shot 63 in the first round in 2013 and finished T-19. A year later, Martin Kaymer opened with 9-under 63 and ultimately won The Players by holding off Jordan Spieth, who has since not made the cut at The Players.
In 2016, two players shot 63, and one went on to win. Jason Day opened with 63 and ultimately won, while Colt Knost also shot 63 in the second round. Knost finished tied for third place on a weekend in which the Saturday round found TPC Sawgrass borderline unplayable because of the speed of greens and some spottiness on the putting surfaces.
In 2018, two players shot 63s, with winner Webb Simpson shooting 63 in the second round, followed by Brooks Koepka in the final round, which included an albatross 2 on the par-5 16th.
Of the six players to shoot 63 at TPC Sawgrass during The Players, four have gone on to win. It's a great sign in a field as deep as this one. A 63 means, aside from Roberto Castro, a player is pretty much a lock to finish inside the top three that year.