Ionic liquids are emerging as promising new electrolytes for supercapacitors. While their higher operating voltages allow the storage of more energy than organic electrolytes, they cannot currently compete in terms of power performance. More fundamental studies of the mechanism and dynamics of charge storage are required to facilitate the development and application of these materials. Here we demonstrate the application of nuclear magnetic resonance spectroscopy to study the structure and dynamics of ionic liquids confined in porous carbon electrodes. The measurements reveal that ionic liquids spontaneously wet the carbon micropores in the absence of any applied potential and that on application of a potential supercapacitor charging takes place by adsorption of counterions and desorption of co-ions from the pores. We find that adsorption and desorption of anions surprisingly plays a more dominant role than that of the cations. Having elucidated the charging mechanism, we go on to study the factors that affect the rate of ionic diffusion in the carbon micropores in an effort to understand supercapacitor charging dynamics. We show that the line shape of the resonance arising from adsorbed ions is a sensitive probe of their effective diffusion rate, which is found to depend on the ionic liquid studied, as well as the presence of any solvent additives. Taken as whole, our NMR measurements allow us to rationalize the power performances of different electrolytes in supercapacitors.