Recovery mode is a Logical separation on your Android Device’s Internal Memory. The Recovery mode serves, as the name suggests, recovering your device to factory settings. However, a custom recovery does a lot more than just restoring your device. It call perform certain low level operations that can be used to flash custom ROMs, MODs, Backup and restore your device, wipe cache, etc.
In order to obtain root access to your HTC One you need to flash a custom Recovery. Here is a short tutorial to flash ClockworkMod Touch Custom Recovery on your HTC One.