In a conventional solar pv system the house will use all the electricity produced from the panels before it imports electricity from the gird. If the panels produce more electricity than the house is using at that point in time it will export it to the grid to be taken up by other users.
In a battery storage solar pv system, this extra power will be diverted to the battery bank in your house and stored to use later when your house requires more energy than the array can provide and at night when the solar pv is not generating.
The wiring of the system is such that the house will use the battery power preferentially ahead of the grid (like conventional solar), thus saving you money as you no longer need to buy as much expensive electricity from the grid. You can use your free electricity that you have already generated instead!
If the battery bank is full, and your house demand is being met by the solar, any excess generation will still be exported to the grid, as usual, so other users can benefit.