The main reason why we don't want our batteries to be overcharged is simply because it drastically shortens their lifespan. The voltage of a full battery under continuous charge gets too high and the water inside starts to separate. In some situations it can lead to the explosion of oxygen and hydrogen inside of the battery. In other cases high voltage might cause the malfunction of home appliances or provoke the inverter to shut the system down.
At the same time, batteries don't like to be used when they are half-empty. Yes, deep-cycle batteries
that are used in PV modules are designed to be deeply discharged, but it doesn't mean that it's healthy for them. This is why the charge controller disconnects the battery when it's half-empty so it can charge up back again and then continue working. The way that controllers track the load of the battery is by measuring its voltage. A fully charged battery has higher voltage than a half-empty one. So when the voltage of the battery drops below a certain level, the controller disconnects it. This function is called Low Voltage Disconnect (LVD) and is relatively modern — nowadays almost all models have it.