DIAGNOSIS This almost always means the thermostat isn't getting power from your HVAC system's wires. The most common culprit is forgetting to connect the 'C' wire, which is the continuous 24V power return.
FIX First, shut off the power to your furnace/air handler at the breaker. Carefully pull the thermostat off its wall plate. Look at the bundle of wires. You need to find an unused wire tucked back in the wall. If you see a blue or black wire not connected to anything, that's your candidate. 1) At the thermostat, strip a little insulation off that wire and secure it into the 'C' terminal on the plate. 2) Go to your furnace. Open the access panel (power is OFF!). Find the control board where the other thermostat wires are connected. 3) Connect that same blue or black wire to the 'C' terminal on the furnace board. Close everything up, restore power, and the Ecobee should boot.
IF THAT DIDN'T WORK If you don't have an extra wire, you might need a PEK (Power Extender Kit). The Ecobee Enhanced should have come with one in the box. It lets you use the 'G' (fan) wire to provide power instead. Follow the included PEK wiring diagram carefully at both the thermostat and the furnace.
HEADS UP When you're working at the furnace, make absolutely sure the 120V power is off at the breaker. The low-voltage terminals are safe once the power is off, but you don't want to brush against the line voltage coming into the unit.