Both appliances together demanded more current than the service wires to the plug could handle without blowing the circuit breaker out. So there was a voltage drop to the microwave. Lower available voltage/more current demanded.

This probably created some overheating of internal components. The average microwave draws 1000 watts or ten amps. A toaster perhaps 750 watts or 7.5 amps.

The cheapest fix is to fish a 12/2 wire to the plug and increase the breaker to a twenty amp one with this change. Or make sure that the two appliances are never used simulatainiously on the same circuit.

This is only sane though if the other plugs on the same circuit are separated as you do not want any remaining 14/2 on a twenty amp circuit. Today the plug by code for the microwave alone is separate. A fifteen amp circuit can service a microwave. Also refrigerators by current code have to be on a separate dedicated circuit as well.

Plugs on kitchen counters have had various code changes required over the years. If the present situation is continued the microwave might eventually be damaged. I suspect you might have triggered some self recovering thermal device in the microwave.