Use the formula $\omega=\sum_\alpha (f_\alpha \circ \pi) \omega_\alpha$ where $\pi$ is the tangent bundle projection $TM \to M$. Connections are defined on $TM$.
Edit: The last sentence should probably read: Principal bundle connections are mappings defined on $TM$. To clarify, the definition I am using here is the following: On the vector bundle $TE$ consider the action of the group $G$ which is induced by the fibre-wise action of $G$ on $E$. Then $TE/G$ is a vector bundle over $M$ in a natural way, and the principal bundle projection (let's denote this by $\pi_E$) has a derivative $d\pi_E$ which is well-defined on $TE/G$. Then a connection is just a right-inverse of $d\pi_E: TE/G \to TM$ (in the category of vector bundles over $M$). So to construct a connection on $E\to M$, you use connections on the trivial bundles $E_{U_\alpha}\to U_\alpha$, forming a weighted sum of them with weights given by a partition of unity $f_\alpha$. Since the connections are mappings defined on $TU_\alpha$ and $f_\alpha$ is defined on $M$ you have to apply the tangent bundle projection $\pi: TM\to M$ (not $\pi_E$) before applying $f_\alpha$. That's how I interpret the formula.