Why will the Hoover Dam stop working before Lake Mead is empty?

The short answer:

Under normal operating conditions, water from Lake Mead flows through Hoover Dam, entering through structures in the intake towers and into penstocks (tunnels in the dam). Electricity is generated as the water spins turbines before exiting out the bottom of the dam.

If the water levels get too low, the turbines have to be turned off because they cannot operate safely (dangerous vibrations occur which can damage the turbines). This water level is called Minimum Power Pool. Water can still be released from the dam, but power can’t be generated from the releases.

If the water gets even lower, it can reach Dead Pool. At Dead Pool, the water level is below all intake structures. There is still water in the reservoir, but the flows downstream will be zero because there is no way for water to make it past the dam.