Instinct is a quick little drive which helps you play up the monster.
For example, a monster with instinct "to feast on flesh" will react differently to a retreating party than one with "to be left alone in it's next" (the latter will probably not pursue, while the former will).
It's just a shorthand reminder at the table.
As far as moves, when you apply them is exactly the same as any other move. As far as how, it's simply fictional... so, you have the monster do what it says! This might trigger other rules or moves, of course (usually, but not always, damage).
Maybe post a few monsters and I can write up examples?
Either from the book or your own creations.
- Alex