Trautman actually proves this statement right after he states it! Let me translate it into perhaps a bit clearer language:
Let $\pi : P = FM \to M$ be our frame bundle and $g: P\to P$ be a small (=connected to the identity) gauge transformation, i.e. $g^\ast \theta = \theta$ where $\theta$ is the solder form. By definition of a bundle automorphism, there is an associated small diffeomorphism $g_M : M\to M$ of the base. Since $g_M$ is small, there is a vector field $X_g$ on $M$ that generates $g_M$. We get a $\mathbb{R}^n$-valued vector field $\tilde{X}^\mu_g$ on $P$ by
$$ \tilde{X}^\mu_g(p) = p^\mu(X_g) $$
where $p\in P, x = \pi(x)$ and $p^\mu$ the frame at $T_x M$ defined by $p$ so that $p^\mu(X)$ is the $\mu$-th component of $X$ in the frame $p^\mu$. The solder form $\theta^\mu$ and the connection form (choose an arbitrary one) ${\omega^\mu}_\nu$ form a basis for the 1-forms on $P$, and hence there is a dual basis $t_\mu, {o_\mu}^\nu$ of vector fields on $P$. Since $g$ is small, there is a generating vector field $Z_g$ on $P$, and we have
$$\tilde{X}^\mu_g(p) = p^\mu(X_g(p)) = \theta^\mu(Z_g)(p)$$
for a vector field $Z$ on $P$ that we can expand as
$$ Z = \tilde{X}_g^\mu t_\mu + {Y^\mu}_\nu{o_\mu}^\nu$$
Trautman then does a computation for the condition that the solder form is preserved that results in
$$ {Y^\mu}_\nu = \nabla_\nu \tilde{X}_g^\mu + {Q^\mu}_{\rho\nu}\tilde{X}_g^\rho,$$
where $Q$ is the torsion tensor, i.e. shows that the condition that the solder form is preserved gives a formula for $Z_g$ in terms of $X_g$. This means there is a map $\mathrm{Diff}_0(M)\to \mathscr{G}$, given by mapping the diffeomorphism generated by $X_g$ to the diffeomorphism generated by $Z_g$, and this map is surjective by construction. Injectivity follows just by noting that different $X_g$ will result in different $\tilde{X}_g^\mu$, which is little more than the statement "different vectors have at least one different component".
