The CO2 actually dissolves into the beer, much like sugar or salt dissolve into water, but it takes some time. The carb chart tells you the final volumes you will reach if given enough time. It takes a week or two with the CO2 maintained at the indicated pressure.
CO2 transfers into the beer at the surface where gas and liquid touch each other. Increasing how much they touch speeds up the process. You can speed up the process by shaking or rolling the keg. Bubbling the CO2 through the beer by turning the keg upside down won’t help much.
Setting the pressure higher than what the chart says does help to force the CO2 into the beer a bit faster but that will risk over-carbonating. The only way to over-carb the beer is to set the pressure higher than what the carb chart says for the given temperature. At the indicated temp and pressure for the desired volumes, you can not overcarb, no matter how much you shake, rattle and roll the keg.