In the John Lee's book Introduction to smooth manifolds there is an unlcear equation with three equalities (-2nd displayed formula on page 69)
$$\dot{\gamma}(t_0)f=d\gamma(\frac{d}{dt}\Bigg|_{t_0})f=\frac{d}{dt}\Bigg|_{t_0}(f\circ\gamma)$$ where I don't know how the first two equalities follow. Can someone give me a hint about the first and second =?
To look up the paragraph in the book, type
john lee smooth 69 "tangent vector acts on functions"
into books.google.com
Sure. For the first, $\gamma\colon (-t_0-\epsilon,t_0+\epsilon)\to M$ is a curve, and by definition $\dot\gamma(t_0)$ is the tangent vector to this curve at $t=t_0$, which is $d\gamma_{t_0}\big(\frac d{dt}\big)$ (i.e., the image under $d\gamma_{t_0}$ of the standard basis vector for $\Bbb R = T_{t_0}\Bbb R$.
The second is exactly how we interpret tangent vectors as differentiation along curves. If $v$ is the tangent vector to the curve $\gamma$ at $t=t_0$, then $v(f) = \frac d{dt}\big|_{t_0} f(\gamma(t)) = (f\circ\gamma)'(t_0)$.