Native After Effects does not provide a way for you to see keyframes for individual points. You would need a third party tool like BAO Mask Avenger.
Or you could create a similar look by using three instances of the Beam effect with their end points attached to nulls via expressions.
which attribute I am altering
Well, you are altering the path shape, which is a compound property that doesn't expose any of its internal data. What Szalam said would be a possible way of doing it, but of course the number of points that can be controlled using these alternate methods will always be limited just as some of those tricks don't cater for closed area shapes.