The monarch draws an extra card at the beginning of their end step, and loses the crown to whoever deals combat damage to them.
The rule
The monarch is a designation, not a card or a counter. A card makes a player the monarch; from then on there is exactly one monarch in the game (CR 725.1 to 725.4).
At the beginning of the monarch's end step, the monarch draws a card. Whenever a creature deals combat damage to the monarch, that creature's controller becomes the monarch.
If the monarch leaves the game, the active player becomes the monarch.
In Life Counter
The crown lies on the table as a token. Drag it to the player who takes it. When the monarch is out of the game, the app hands the crown to the active player, as the rule says.
