Tangent space of a manifold

Question

For a manifold, $M$, at point $p$ we have a set of real-valued smooth function, $C^\infty_p(M)$ passing through $p$. To prove the tangent space at point $p$ we assume that we have map $C^\infty_p(M) =\{f:M \to R\}$, $\gamma:R \to M$ and try to find $\frac{df(\gamma(\tau))}{d\tau}$. The map $\gamma$ takes a point $\tau$ from one-dimensional curve and give a point on the manifold along the curve $\gamma(\tau)$ where $\gamma(\tau=0) = p$. But I can not understand what kind of mapping the function $f$ does? Is it take any one of the curve passing through $p$ and give a real value? Then what direactional derivative $\frac{df(\gamma(\tau))}{d(\tau)}$ mean in that case? Is it the rate of change of the curve when I change the parameter $\tau$ slightly?