What does Hot Boot mean?
If it works like other "hot boot" apps and functions, then it basically restarts the Android GUI processes/threads. There was an XDA news post a while back after someone made an app for it where the app author described it as such:


  Hot reboot by rebooting only the graphical Android shell.


From what I know it's sort of equivalent to restarting your X server in a Linux environment (like hitting CTRL + ALT + BACKSPACE, for example). I would definitely doubt that you'd cause any damage by restarting your graphical environment, especially considering other OSes do this pretty easily, but I suppose there's an outside chance if MIUI implements it strangely.

Typically the reason people use a "hot boot" option is if the UI has somehow gotten sluggish and would benefit from a restart but they want to save time by not restarting the whole system (maybe a memory leak in the UI or something of that sort). It could also be useful if you need to restart the UI to apply some sort of change, like a theme perhaps.