Force carbonation dissolves CO2 directly into cold beer under regulator-controlled pressure — typically in a keg — reaching a target carbonation level through gas pressure and temperature rather than through residual yeast fermenting a measured sugar addition the way bottle conditioning works.
Cold beer holds more CO2, and that's the whole mechanism
Colder liquid dissolves more gas at a given pressure than warmer liquid does, which is why force carbonation setups specify both a target pressure and a target temperature together — a carbonation chart isn't giving a pressure number alone, it's giving a pressure-and-temperature pair, and changing one without adjusting the other moves the beer away from the intended carbonation level.
Two common methods, one faster and rougher, one slower and gentler
Setting a keg to serving pressure and simply waiting (roughly a week to fully equilibrate) is the gentlest, most predictable method; shaking or rocking a keg at higher pressure speeds equilibration to hours instead of days but is easier to overshoot if not monitored carefully. Both reach the same destination — a specific dissolved CO2 volume at a specific temperature — by different routes, and the choice mostly comes down to how much patience a brewer has on a given day and how much risk of overshooting they're willing to accept.