Let us show that the integral converges absolutely, if $|z|<R$:
$$\int_0^\infty |F(tz)e^{-t}| dt \le \int_0^\infty \sum_{n=0}^\infty \frac{|a_n z^n|}{n!} t^n e^{-t} dt = \sum_{n=0}^\infty \frac{|a_n z^n|}{n!} \int_0^\infty t^n e^{-t} dt.$$
Note that the interchange of summation and integral is justified by Fubini-Tonelli (since the integrand is non-negative). Also please take care to understand that in case of divergence (which happens if $|z|>R$), this calculation still makes perfect sense, by positivity. That is, in case of divergence, all the terms simply equal $\infty$.
Now we recognize the Gamma function,
$$\int_0^\infty t^n e^{-t} dt =: \Gamma(n+1) = n!,$$
which can be verified using repeated integration by parts.
Thus the sum on the right in the previous display equals
$$\sum_{n=0}^\infty |a_n z^n|$$
which converges to a finite value by assumption since $|z|<R$.
Also you see, that to prove your claimed relation $f(z)=\int_0^\infty F(tz) e^{-t} dt$, you just need to follow this argument without the absolute values.