There is a general theorem by Karass, Pietrowski and Solitar, in their work "Finite and infinite cyclic extensions of free groups".
J. Austral. Math. Soc. 16 (1973), 458–466.
The theorem says, in the language of graphs of groups, that for any finite graph of finite groups, its Bass-Serre fundamental group has a free subgroup of finite index.
Your example can be expressed as a finite graph of finite groups in the following manner. Take a graph $\Gamma$ with one vertex $V_0$ of valence $m+n$, $n$ vertices $V_1,...,V_n$ of valence $1$, $m$ loop edges based at $V_0$, and $n$ edges connecting $V_0$ to $V_1,...,V_n$ respectively. Each edge is labelled by the trivial group, $V_0$ is labelled by the trivial group, and $V_1,...,V_n$ are labelled by the groups $G_1,...,G_n$ respectively.
You can think of the subgraph of loop edges as having fundamental group $F_m$, and the subgraph consisting of the edge connecting $V_i$ to $V_0$ as having fundamental group $G_i$. By identifying the $V_0$ points of all of those subgraphs one is taking the free product of all of those pieces.