The Seiko Museum Ginza on why it mattered: "The invention that replaced weights as a driving force was the mainspring. Compared to a weight, a mainspring was much smaller and not so easily affected by movement or tilting of the clock." A weight-driven clock must hang still; a spring-driven one can be carried.
How its energy reaches the hands, per Grand Seiko: "A wound-up mainspring exerts force to turn gears at a set speed as it unwinds… the mainspring unwinds gradually and evenly over a long period of time, and this consistent motion is transmitted through the gear train."
In an automatic watch the mainspring is wound by the wearer's motion turning a rotor, and by the crown. Neither page cited here states that in those words, so nothing is quoted for it; the automatic and rotor entries carry the sourced account.
Not stated here: the barrel that houses the mainspring. No Tier 1 or Tier 2 source reachable for this entry describes it, so it is not described.