Vector fields - Chapter 0, Do Carmo's Riemannian Geometry

The meaning of the notation (though it is clumsily written) is that of taking the derivative of $f$ at the value $\varphi(p)$ with respect to the vector $d\varphi_p(v)$ since in this case $v\in T_pM$ and therefore $d\varphi_p(v)\in T_{\varphi(p)}M$. A much clearer way to convey the same idea would be to write $\left. \left (d\varphi_p(v)f\right ) \right |_{\varphi(p)}$.

I've worked through a lot of do Carmo, and I can comfortably say as a fair warning that his notations are less clear than other treatments of manifold theory and Riemannian geometry. In my experience, do Carmo's chapter 0 is a poor place to learn the subject for the first time. I'd highly recommend comparing his text to analogous statements in other books. For example, consider using John Lee's Introduction to Smooth Manifolds as a companion text. It might be easier to come back to do Carmo and learn Riemannian geometry there after learning about manifolds elsewhere.