The breaker limits the current to the whole circuit. The breaker actually reacts to the heat in the breaker. Higher current in the breaker creates more heat in the breaker.
The rules about minimum wire size allowed is evaluation of the ability of the insulation to not melt under maximum load. Larger wire creates less heat when current flows through it.
Without taking a full hour and a whiteboard and a bunch of math teaching the relationship of voltage/resistance/current/watts/load/heat the important factor is that wire has resistance, resistance reduces the voltage available at the receptacle. Bigger wire has less resistance, and less voltage is lost on the wire feeding the load. Voltage lost on the wire is directly related to heat, 3.4 btu's per watt/hour.
So for a theoretical imaginary 15A load on a circuit with #14 wire that is +/-100' from a breaker the voltage available at the end of the wire might be reduced from 120v to 116v, the heat created on the wire would be around 60 watts, or around 200 btu's. If the circuit was installed with #12 wire the voltage might be 118v, only creating 100 btu's in the walls.