Smooth atlas is defined to be a collection of charts that cover manifold and any two charts in atlas are smoothly compatible. $f$ is smooth if for every chart $(U,\varphi)$ composition function $f \circ \varphi^{-1}$ is smooth as a map from $R^n$ to $R^k$.
In fourth sentence of the proof it is written that $(U,\varphi)$ is smoothly compatible with $(U_\alpha,\varphi_\alpha)$.
Why?
$(U,\varphi)$ is not necessarily in atlas so why is it smoothly compatible with chart from atlas?
Remember that the "smooth structure" on $M$ is a maximal smooth atlas. So even though $\{(U_\alpha, \phi_\alpha)\}$ is an atlas, it does not necessarily contain every smooth map. However, given any smooth atlas, there is always exactly one smooth structure containing it--we could call this structure the structure "generated" by your atlas. The structure is precisely the set of all charts which are smoothly compatible with the generating atlas.
This is what is meant when the proof takes an arbitrary smooth chart $(U,\phi)$, that it is an arbitrary element of the smooth structure. So of course it is smoothly compatible with each $(U_\alpha,\phi_\alpha)$.