Once the engine is running, the battery serves the same role as your towns water tower. It is a resivor of energy, for times when demand exceeds the rate that the alternator can supply.
Turn on every electrical device in your plane, radios transmiting, lights, pitot heat, ect. You are now drawing 55 amps per hour out of your 40 amp per hour altrnator. Where are the extra 15 amps coming from? Answer, your battery. If your battery has 65 amps, it can supply this load for 4 hours.
Once you turn off the landing light, and stop transmiting over the radio, you are only useing 35 amps, and 5 amps per hour are returned to the battery.
Same principle with a water tower, at half time, everybody goes to the bathroom, creating a short period of high demand. The water stored in the tower under pressure, drains and meets this demand. Durring the second half, the pumps refill the tower.
Power spikes useually refer to voltage spikes, which is the voltage regulators job. Think of a pressure wave inside a hose, because somebody parked a truck on it. If there is a weak spot in the hose, then it can burst.
The battery absorbs "spikes" in the draw of Amps, like the water tower.