I think what the XBox is doing is that it is running in a low power mode when idle, and is able to detect when the TV selects its HDMI port as the source. When it detects that, it boots the full XBox operating system, and the fan and LEDs turns on. You want the same functionality with the Raspberry Pi.
As far as I know, there is no readily available software to do exactly that. I think most people just leave their Raspberry Pis on, since it is a fairly low power device by default.
There are a few questions here that try to optimize power usage (e.g. How to achieve Low Power Consumption on Raspberry Pi via Software and Hardware control), and perhaps it would be possible to configure the Pi to lower its power usage slightly when idle, and then re-enable things like USB ports when the TV selects its HDMI port (presumably it would be detected somehow with HDMI-CEC). I am not aware of any existing software that does this.