Pull off just the tiniest amount of black fibre and roll it between your fingers until you have a tiny ball the size you want for eyes. Roll both eye balls before attaching them to make sure they are the same size. Place the eyes where you want them and felt onto the face.
Tip: when attaching a tiny piece, always felt around the edges of it first - if you felt into the middle it can poke right through and disappear. If your owl body is not felted hard enough it can be hard to attach tiny details - if this happens just felt your owl some more all around. If your eye becomes too small when felting it on just roll another bit of wool into a ball and felt it on top of the first bit. Not too much happens in needle felting that can't be fixed!
I hope you have enjoyed needle-felting your owl. Now that you have the basic techniques you can keep felting as far as your imagination will take you.