Kerethos said:
It's been "the standard" as it were since the prevalence of LCD monitors, about a decade or so give or take. CRT monitors generally had higher refresh rates, anywhere from 75-120Hz+ typically (some went much higher). LCD screens however took up less space, had flat screens, etc but lower refresh rates.
Most LCDs, even today, are around 60Hz. Some are 75Hz and there are current (TN only) screen models that refresh at 120Hz (this is mostly so that they can show 3D stuff at 60Hz per eye, though 2D @ 120Hz is also a bonus). The reason for wanting 60FPS in the game is so that it matches the screen's refresh rate. If matched perfectly, a game will look and play silk smoothly with no screen tearing or lag. Lower frame rates means some frames have to be duplicated. Some games go so far as to "step down" completely from 60 -> 30 if 60 can't be maintained as then each frame can be doubled and play can continue, albeit (arguably) less smoothly.
At an in between number of frames, the game will look disjointed and there will be frequent screen tearing. Triple buffering and VSync both help in that regard. In an ideal world, a game's FPS would precisely match the refresh rate of its screen. Since consoles generally can't maintain 60fps, they are stepped down to 30fps with each frame doubled for 60Hz screens (the most common kind). Just know that if one's screen had a 75Hz refresh rate, 60FPS games would also suffer screen tearing and artefacts during fast scenes, so one would want to play at 75FPS ideally. I can only dream of how great games would play at 120FPS on a suitable screen (albeit a TN display), but my most recent screen is a 60Hz IPS monitor.
Last point of note is that it's only really consoles which lock frame rates in games since PC frame rates are essentially "unlimited" and variable, dependant on the power of the PC/graphics card (and obviously the monitors ability to display it). Hence we get graphical settings to tweak for the right balance of visuals vs. performance with the top end cards generally powerful enough to max out everything and still have the grunt to maintain high FPS.