How do floating gate transistors not leak the stored charge?
The short answer is that they do not store the charge indefinitely, although the data retention time is in decades, usually.

Floating gate devices are charged across a quantum tunnelling oxide at a relatively high voltage. 
In simple terms. a high voltage is applied to the control gate, and electrons travel across the tunnel oxide to the floating gate. There are a number of subtleties I have not dealt with here, but the basic idea is correct.

In modern parts the charge pump is within the IC, whereas the older crowd will remember devices that had a 'Vpp' pin.

The tunnel oxide is very thin, incidentally. The process node for a device may be 65nm (about the smallest practical node for NOR flash, incidentally) but the tunnel oxide is typically 5nm to 10nm thick.

The charge on the floating gate eventually discharges to the un-programmed state the same way it was programmed; by quantum tunnelling.

Note that floating gate devices are highly susceptible to discharge under X-Ray (a commonly used process in manufacturing to inspect solder joints, particularly under BGA devices), so it is prudent to have in-system programming capability.

HTH