Storage heaters use off-peak electricity which is supplied from the national grid to your house, usually overnight when demand for electricity nationally is low. This is called the 'off-peak' period
Because it's off peak, it's supplied at a cheaper price than standard rate electricity, so it needs a separate off-peak electrical circuit and meter. This circuit is dedicated to operating the off-peak heaters and is only switched on during the off-peak time period. This will take place at certain times of the day or night, and will be dictated by your electricity supplier.
In the same way that your kettle uses an element to heat water, electricity is used to heat elements in your heater. Over a number of hours, the elements gradually transfer the heat to a very high-density material that absorbs and stores the heat for use the next day. The off-peak heaters use insulation material to retain as much of this heat for as long as possible.
When the off-peak period finishes, for the majority of off-peak heaters, the heat is gradually released into the room in a controlled way over the course of the day. In the case of smart fan-assisted models almost all the heat is retained until it is required.