Batteries store electrical energy as chemical potential energy. When the battery discharges, a chemical reaction is taking place inside the cell that causes electricity to flow.
Charging a battery involves forcing that same chemical reaction to happen in reverse. Once all of the compounds available to be reacted have done so, the battery is full. Any extra energy will do something other than charge the battery, so the cell might start to heat up, the terminal voltage might rise rapidly, electrolyte might disassociate, or any number of other bad things.
In the old days, you could just apply a fixed voltage to a lead-acid battery and it would gradually stop charging. This wasn’t the optimal way to manage the charging process even for a simple battery chemistry, but it was often ‘good enough’.
With high performance lithium chemistry batteries, there’s essentially a small computer (microcontroller) that monitors the state of the battery at all times. When that Battery Management System determines that the battery is fully charged, it essentially disconnects the battery from the power supply (often called a charger). It will do the same thing if it detects a fault scenario, for example if the battery is too hot to be safely charged.
TLDR; there is dedicated circuitry that monitors modern batteries and disconnects the flow of charging electricity when it determines that the battery is full.
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