Breakers trip when an overcurrent condition occurs. That's it.
With some caveats we can ignore, overcurrent = load greater than the breaker's rating. For example, if you draw 50A on a 15A breaker, it will trip.
However, overcurrents can be considered in two groups.
Some are effectively infinite - an infinite draw on a breaker of any size is always an overcurrent. This happens when there's a short circuit, for example if a rodent chews through the hot wire in the wall, and it touches ground. If you're getting a short circuit in your home wiring, a repair will be required.
The other kind is typically within the same order of magnitude of the breaker's rating, for example if you put three tea kettles on the same circuit in your kitchen. You want the breaker to trip here, too, but it's annoying. This is the user's error, and can be remedied by putting devices on separate circuits, or only running 1 at a time, etc.
In all cases, the purpose of the breaker is to protect the in-wall wiring for that circuit. If the breaker did not trip, then the wire would act as a fuse, heating up, and eventually melting to break the circuit. A fire might have started in the meantime.
Usually a breaker is at the beginning of a circuit, but the main breaker for a panel (or subpanel) is often at the end of the circuit. That means that flipping the breaker off will not de-energize the wire it is protecting.