DIAGNOSIS This is a classic misunderstanding of system types. Your old thermostat was battery-powered and only needed tiny signal wires (R and W). The Nest E needs constant power, which usually comes from the 'C' (common) wire. Your system is sending power on 'Rh' for heating, but without a 'C' wire, the Nest is struggling to get enough juice to run its brain, causing the error.
FIX The proper solution is to install a 'C' wire. Look at the bundle of wires coming from your wall. If there's an unused wire (often blue or black), you're in luck. 1) Turn off power to your furnace. 2) Connect that unused wire to the 'C' terminal on the Nest base. 3) At your furnace control board, find the 'C' terminal (often labeled 'COM' or '24VAC COM'), and connect the other end of that same wire there. This gives the Nest a complete 24V circuit to sip from.
IF THAT DIDN'T WORK If you don't have an extra wire, you have a couple of workarounds. You can install a Nest Power Connector at your furnace, which is a little module that provides the 'C' wire function without running new wire. Alternatively, you can try using the 'G' (fan) wire as a temporary 'C' wire, but this will disable your ability to run the fan independently from the app.
HEADS UP Running a Nest without a 'C' wire using its "power stealing" feature works for many, but not all, systems. In some cases, especially with heat pumps or systems with an electronic air cleaner, it can cause the clicking issue, Wi-Fi drops, or even cause your system to short cycle. A 'C' wire is always the recommended, stable setup.