I found this statement from my text very confusing:
What does it mean by identification of each tangent space $T_pV$ with $V$ itself? - what does "identification" really mean here?
If it means isomorphic, then it conflicts with my understanding that each tangent space $T_pV$ is locally isomorphic to $V$.
What is $Xf$? I don't understand the expression. I don't know what is the function $f$.
Thank you very much for your help.
What is implicit in the author's comment is the view of a tangent vector as a linear functional on the space of smooth functions defined near $p$.
This is a somewhat roundabout way of thinking of a tangent vector, and it does not connect easily with intuition, but it can be efficient when you study smooth manifolds.
The author's point is that when the manifold happens to be a vector space $V$, there is a natural choice for such a functional, namely the one given by the formula $Xf = \frac{d}{dt}\big\vert_{t=0}$, etc. In this case the roundabout approach is clearly redundant, but it needs to be shown that it is consistent with what we expect it to be intuitively, namely the directional derivative.