Depends on the appliance.
Light bulbs/heaters, it’s literally just electrons flowing through a wire and creating light/heat because the material has a resistance to them flowing through.
Anything that moves/has a motor makes use of the fact that alternating electron flow creates a changing magnetic field, which then moves stuff if configured properly.
Anything electronic (vs electrical) is much more complicated; depending on the device voltage or current, which are properties of electricity, are interpreted as signals that is then processed with e.g. a circuit board. Applications can range from radios to computers and sensors.
Generally the “why” gets very mathematical and theoretical, often using models that work but can’t really be proven. At one point you even just have to accept that some phenomena are basic rules of the universe
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