Airport security officers may, randomly, ask that you turn on electronic devices (usually do something to activate the display) to show that they are functional. To do this, the battery must not be completely discharged. If you are unable to turn on the device, they may not allow you to bring the electronic device through the checkpoint and you would have to dispose of it or otherwise arrange to get it to your destination (or, perhaps, find a charger).
This related to a US TSA program, as discussed by the TSA in this press release, this blog post, and this BBC article. While the TSA does not screen passengers outside the US, it does work with the authorities in other countries. Other countries do perform similar checks as part of their security program, even for flights not involving the US, or if they detect anomalies on an X-ray. The UK, for example, advises that electronic device batteries should be charged for all flights in or out of the UK.
This is not an entirely new procedure. Asking travelers to briefly turn on laptops was something that occasionally happened even pre-9/11.
In my experience, such checks are rare, but can happen randomly. I've been asked by the TSA in the US and once at a pre-boarding check on SWISS in Zurich flying into the US.