how it handles the load exceeding capacity

As in what happens if you plug too much stuff that it exceeds your solar production?

I'll use mine as an example, but it might be different with different models and configurations:

Inverter can handle up to 10kw
If solar production is at 5kw, and home is demanding 7kw, in my case, I have it set up as to draw the remaining 2kw from the battery, if battery is depleted, it will draw 2kw from the utility company

If home demands more than 10kw that the inverter can handle, it will trip the internal inverter protection or a circuit breaker leading to it