Had my start in automotive electrical work, DC, generator, starters, alternators. Old school points and condensers, condensers would take a large charge quickly from the coil, and dissipate quickly as points closed to create a spark for engine to run. Very high voltage, low amperage, have in neighborhood of 16,000 to 30,000 V, HEI would develop some in excess of 80,000V. Unless AC voltage capacitor is totally different than DC. Here is one simplified source quote about capacitors:"Hard start" kits don't exactly store energy to be used to start the motor. They are simply a higher value capacitor than the run capacitor already used in the motor and a potential relay to switch it out of the circuit after motor reaches certain conditions. The phase angle between main and aux windings remains the same. The added capacitance will increase the current in the aux winding and that is what gives the starting boost. It is not a soft starting method, in fact sort of the opposite. It will get the motor started more quickly presuming the supply circuit can deliver what is demanded.
When used on a typical AC compressor they turn what was a PSC motor into a CSCR motor. If the supply is marginal at being able to deliver the starting current needed, they may still help get the motor started, but is still going to draw high current when trying to start. A true "soft starting" method would ramp the current up as motor speed increases.
"A capacitor (originally known as a condenser) is a passive two-terminal electrical component used to store energy electrostatically in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors (plates) separated by a dielectric (i.e., insulator)."