Diagnosis
The device failed to download or apply a system software update correctly. Corrupted data is blocking the boot process.
How to Fix It
You need to force a fresh update via recovery mode. Unplug the Chromecast. Hold down the button on the side of the device. While holding it, plug the power back in. Keep holding until you see the recovery screen on your TV. Use the remote to select 'Apply update from ADB'. Then select 'Reboot system now'. The device will attempt to download the update again.
If That Didn't Work
In the same recovery mode menu, select 'Wipe data/factory reset'. This erases everything and installs the original OS version, which will then update normally.
Heads Up
This process requires a stable internet connection. Use a strong Wi-Fi signal or connect an Ethernet adapter to the Chromecast's USB-C port before starting.