On page 78 of Hatcher here he constructs an example of the universal covering space of $\Bbb{R}P^2 \vee \Bbb{R}P^2$. Now from the picture in example 1.48 I think I understand why it is a covering space. Our covering space map $p$ is the usual quotient map from $S^2 \to S^2/\sim$ when we look "locally" around a point on say the sphere with equator $b^2$. I also get that this space is simply connected because it is just the wedge sum of a bunch of $S^2$'s that has trivial fundamental group by the Van Kampen Theorem.
Now my problem is my understanding of this and the usual construction of the universal cover as consisting of points that are homotopy classes of paths is out of sync. For example, in the picture given in Hatcher I don't get why say the point $ba$ would represent the homotopy class of some path in $\Bbb{R}P^2 \vee \Bbb{R}P^2$. Can someone explain on this a little? We just learned about orbit spaces and covering space actions today and they are a little confusing.
Thanks.