I'm finding in various questions/textbooks that for Bell number $B_n$, defined as $$B_n = \sum_{k\geq 0}\left( \begin{array}{rl}n \\ k \end{array} \right) B_k$$ the exponential generating function is known to be $$B(t) = e^{e^t - 1}$$
I can check that this holds by, i.e., expanding it to Taylor series, but how is this expression derived?