DIAGNOSIS The thermostat isn't getting power from your HVAC system's C-wire or the internal battery is dead. This is a power issue.
FIX Pull the thermostat off its base. Check the connectors on the back of the thermostat and the wires in the wall plate. Ensure the wire labeled 'C' is seated firmly. Reattach the thermostat. If the screen lights up, the connection was loose.
IF THAT DIDN'T WORK Your system may not have a C-wire. The Nest was running on its internal battery, which is now depleted. You need to charge it. Use the Nest's USB cable and plug it into the micro-USB port on the back of the display. Charge it for at least an hour, then reinstall.
HEADS UP If you don't have a C-wire and this happens repeatedly, you'll need to install a C-wire adapter or use a common wire kit. Running without a C-wire will kill the battery over time.