How do appliances prefer to use my solar power from a grid-tie inverter over the power from the grid?

I have a grid-connected solar inverter. In the swtich board the incoming wire from the street, the incoming wire from the inverter and the outgoing wire to the loads in my house are all connected together (wired in parallel).

It appears that the loads in the house will consume all available solar power (if needed) before drawing power from the grid.

How does this work? Why doesn't the load draw power equally from the solar inverter and from the grid?

I suspect that the inverter monitors the grid voltage and produces an output voltage that is just a few volts higher. Is that all that is needed for the loads in the house to use all the solar power before using grid power?

Kirchhoff's current law works for you.

At some point in your installation, the mains supply, the inverter output and the load representing the appliances in your home must all meet. Kirchhoff's law says that the total current going into that point must equal the total current going out.

If your appliances are using more power than the inverter is generating, then it must be true that...

(power from inverter) + (power from grid) = (power to appliances)

If the inverter is generating more than you need, then...

(power from inverter) = (power exported to grid) + (power to appliances)

Either way, as much power as possible is coming from the inverter to the appliances.

I suspect that the inverter monitors the grid voltage and produces an output voltage that is just a few volts higher. Is that all that is needed for the loads in the house to use all the solar power before using grid power?

Yes.

This becomes more obvious if you have a meter between the two: meters measure current, and if the current is flowing out (indicating higher generation than usage), that implies the voltage on the house side is higher than the grid side.

simulate this circuit – Schematic created using CircuitLab

Grid tie inverters monitor the circuit voltage and if it is in an acceptable range produce a current into the supply. they don't try to match the voltage they just produce (phase matched) current.

Since a current can only flow one direction in a piece of wire the inverter's current flows to your appliances first, and if there is any excess it flows out onto the grid, if there is a shortfall it is made up by the grid.