The Discovery Points are a bit similar to experience points from role-playing games. Collecting more Discovery Points provides more powerful spells (actually, I didn't notice anything has changed). Also, you can collect them whenever you want to. However, the main question is: how to find them?
The main rule is simple: you have to stick your nose anywhere you can (and also where you cannot). Talk to the paintings, cast various spells on highlighting items (sometimes you have to repeat it few times!), light the torches, clean up, etc. Pressing C on your keyboard might be helpful too - you will see how many percent of the points you've found in certain area and the items which have to be used in order to gather more points (they pulse in this view).