DIAGNOSIS The thermostat isn't getting any power. On this model, the most common cause is the C-wire (common wire) not being properly connected or providing power at the wall plate.
FIX First, pull the thermostat display straight off its wall plate. Check the bundle of colored wires. You need to see a wire connected to the terminal labeled 'C'. If there isn't one, your original HVAC setup might not have included it. If there IS a wire in the 'C' terminal, ensure it's snapped in securely. Go to your furnace/air handler and open the panel. Find where the thermostat wires connect (usually a similar terminal block). Verify the corresponding wire is firmly attached to the 'C' terminal there too. A loose wire at the furnace is a classic culprit.
IF THAT DIDN'T WORK It's possible the furnace door safety switch isn't fully engaged, cutting power to the thermostat circuit. Make sure the furnace/air handler access panel is clicked all the way shut. Also, check your home's circuit breaker for the furnace/HVAC system and reset it if needed.
HEADS UP If your old thermostat worked without a C-wire, it was probably stealing just enough 'phantom' power. The RTH9585WF needs a dedicated C-wire for consistent power. If you don't have one, you'll likely need to run a new wire or use an add-a-wire adapter kit, which is a whole other project.