Apple - How does an iOS device keep time when it is out of power?

The iPhone is not completely depleted when the battery "dies." If you'll notice, when you try to turn it on when the battery is dead, the screen will come on briefly to tell you the battery is dead -- i.e., there is still some energy left in the battery. The clock continues to run off of the battery, and takes such a small amount of energy that Apple decided to keep the clock going.

Eventually, the battery will completely die (probably after many days or weeks), and the clock will eventually stop keeping time. As Ian C. mentioned, the clock will synchronize when the phone does get plugged in and turned on again, if a network is available.

The phone synchronizes its clock with your wireless provider's embedded clock signal when it starts up. All cellular protocols provide for a clock network feature that lets devices on the network synchronize their time and date with a master time and date from the provider.

So it isn't that the phone kept time while it was out of power, it was that it set the correct date and time when started up.

The Real Time Clock is a component independent of the primary CPU (although it's all part of the same SOC IC) which will continue running from a lower voltage than is required to run any other part of the phone. As a phone's battery is depleted its voltage decreases, and below a certain voltage some components will fail to function at all. It may be depleted so far that it would be considered permanently damaged and un-rechargeable and it would still have enough voltage to run the RTC.