In video games, if you rapidly turn around, you will experience a phenomenon called 'screen tearing'.
Imagine a pole in the game, and if you turn quickly, instead of the entire pole moving past your field of view, only the lower or upper half moves, while the other half stays there for a split second, and then moves away as well.
This phenomenon is caused if your graphic card is working faster than the refresh rate of your monitor.
Turning VSYNC on tells the graphic card to not produce updated imagery (frames) any faster than your monitor's screen rate (usually 60 - 75Hz; 120 Hz if you're a rich kid or still have a CRT, lol)
...thereby "dropping" the frames down to whatever your monitor's maximum display rate is, eliminating the screen tearing.