I don't understand what you don't understand, sorry.

Think of it this way - for every new password that you put into the database, run it through some function that spits out N bits. Remember these bits. Then, when you have a password candidate, pass it through the same function and check if the bits are the same. That's all there's to it.

The main exploitation risk is that hinting can be used to quickly reject candidate passwords when running a brute-force discovery (so that only remaining positives need to be run through the full authentication process). The provisions listed are to mitigate this risk, i.e. to make hinting unfit for the quick pre-filtering purpose.