Ultimately it's just down to the small voltage differences in the interconnecting cables. Every conductor has some resistance, and a current flowing along it will result in a voltage difference between its ends. If the current flows in the opposite direction then the voltage is also reversed (in a.c. that typically means the current is then 180 degrees out of phase with the voltage but that's just the mathematical convention). For a PV system to output a certain power, it need to raise its output voltage until the required power flows. When the power it generates exactly equals that consumed on-site, then the voltage at the connection point due to the PV inverter will exactly match the grid voltage, so no current flows to or from the grid. When the PV inverter produces more power, the voltage will be slightly higher and currents flows back to the grid, lower power and the voltage will be reduced and some current from the grid will make up the deficit.

It's the same kind of thing as "how does a consumer unit know how to route the right amount of current to the individual circuits" - in short, it does nothing clever - it's just down to Ohm's law and things trying to find a natural steady state.

   - Andy,

The solar inverter synchronizes itself to the incoming mains.  Then when the sun comes out, it adjusts its output phase and voltage to ensure it's pushing out as much current as it can.

Then Kirchhoff's Current Law sorts the rest.  That law states that the total current into any node must equal the total current going out.  Which is common sense when you think about it.  One of those nodes will be the junction where the incoming grid mains, the solar output and the supply to the main consumer unit all meet.

C Brown said:

what prevents the DNO supply from supplying power to the house load as well (an thus the DNO charging)

You, as the consumer, matching the load to what is available from the PV and/or battery inverters.

But only the shortfall will be drawn from the grid, because the inverters are connected after the DNO/suppliers meter.

If it helps (it might not) and if you can visualize water flow more easily, think of a tap connected by a length of pipe to a huge reservoir - turn on the tap and water flows (at a given rate) turn it off and it stops. Now add a T piece a bit before the tap with the extra branch being fed by a constant volume pump (supplied by a pond or bore hole or something). Turn the pump on while the tap is off and all it's output goes to the  reservoir. Turn the tap partly on and some of the pumps's output will go to the tap the excess still goes up the pipe to the reservoir. Open the tap fully (so it can draw more water than the pump alone can supply) and it'll take all the pump's output plus some from the reservoir. Water can only flow in one direction at a time in a pipe - similarly electrical current.

    - Andy.