You can also approach a proof of a more general relation using the generalized Dobinski formula:
$$f(\phi.(x))= e^{-x}exp(a.x)=exp(-(1-a.)x),$$
where $(\phi.(x))^n=\phi_n(x)$ is the $n$th Bell polynomial with $B_n=\phi_n(1)$ and $(a.)^n=a_n=f(n).$
Then $$\sum_{k=0}^\infty\phi_k(x) t^k=\frac{1}{1-\phi.(x)t}=e^{-x}\sum_{n=0}^\infty\frac{1}{1-nt}\frac{x^n}{n!}$$ $$=\sum_{n=0}^\infty \frac{x^n}{n!}\sum_{j=0}^n(-1)^{n-j}\binom{n}{j}\frac{1}{1-jt}.$$
And, the last finite difference expression is the partial fraction expansion of $n!\prod_{j=1}^n \frac{t}{1-jt}$, so
$$\sum_{k=0}^\infty\phi_k(x) t^k=\sum_{n=0}^\infty x^n \prod_{j=1}^n \frac{t}{1-jt},$$
which reduces to the illustrated formula when $x=1$.
Other proofs, including those alluded to in other answers, can be found in W. Lang's notes.