DIAGNOSIS The RTH9585WF uses two AAA batteries as a backup, but if it's constantly running on them (or thinks it is), it locks you out to preserve power. The main issue is that the system isn't getting proper 24V power from your C-wire, so it's draining the batteries trying to run the display and Wi-Fi.
FIX This is a power problem, not a battery problem. Pull the thermostat off the wall and remove the batteries. Check the wire in the 'C' terminal on the wall plate. Is it there? Is it snug? Now, with the display off and batteries out, go check your furnace. Ensure the panel door is shut completely (tripping the safety switch) and that the corresponding 'C' wire is secure at the furnace control board. After verifying, reinstall the thermostat display. It should boot using C-wire power. Then, and only then, put fresh AAA batteries in.
IF THAT DIDN'T WORK Use a multimeter to check for voltage between the 'R' and 'C' terminals on the wall plate (with wires attached). You should see a steady 24-28 volts AC. If you see 0V or a very low reading, the problem is at your HVAC system—a blown fuse on the control board is a likely suspect.
HEADS UP The thermostat will always show a 'battery' icon in the status bar, even when running on C-wire power—that's normal. The 'battery low' warning is different. Also, using rechargeable NiMH batteries in this thermostat is not recommended; they have a lower nominal voltage and can trigger low battery warnings prematurely.