Classic pinball machines often have a “Same Player Shoots Again” insert near the flippers or a light-up section on the backglass that says the same.
The convention of winning a free ball and playing that ball immediately goes back to the days of relays and mechanical devices. Early electro-mechanical pinball machines had no memory except for what could be held with a relay.
When the player won an extra ball via points or some other kind of scoring, the ball count unit that keeps track of which ball the player is on, would simply be prevented from advancing when the ball drained.
Typically on a ball drain, the switch under the apron would advance the ball played unit. But in this case the “Same Play Shoots Again” light would light and the ball played (and player unit) would not advance until a switch is hit.
Basically, advancement is suspended and the player gets to plunge again.
This arrangement of extra balls being played during the pinball player’s turn carried on into the solid-state era and with contemporary pinball machines.
And it makes sense. The player doesn’t get any more “turns” up at the machine but rather an extended turn. Playing pinball with someone who racks up a bunch of extra balls and then has an extended time at the end of the game would be kind of like rubbing it in — like one of those dreaded endings in a game of Risk where the winning player simply pummels the loser for the last half hour of the game.