In page 218 of this pdf, the author says if we set $x_i = 1$, we are simply counting the number of partitions of {1,2,3..m} and he says that
$B_{m,k} (1,1,1,1,1..) = mSk$
where $ mSk$ is the Stirling number of second kind
but I don't understand what's the intuition behind this, what exactly happens when we put in values to the polynomial, as in how do we interpret it?
The combinatorial definition of the Bell polynomial $B_{m,k}$ is a sum over partitions of $\{1,2,\dots,m\}$ into $k$ blocks. For each partition of $\{1,2,\dots,m\}$ into $k$ blocks of sizes $j_1, j_2, \dots, j_k$, we include a summand $x_{j_1} x_{j_2} \dotsb x_{j_k}$.
When we set $x_1 = x_2 = \dots = 1$, then each of these summands $x_{j_1} x_{j_2} \dotsb x_{j_k}$ also simplifies to $1$. As a result, $B_{m,k}(1, 1, \dots, 1)$ is also a sum over partitions of $\{1,2,\dots,m\}$ into $k$ blocks, but for each partition, we simply add $1$, regardless of what the sizes of the blocks are.
A sum of many $1$s just simplifies to however many $1$s there are. In this case, it simplifies to the number of partitions of $\{1,2,\dots,m\}$ into $k$ blocks, which is exactly what the Stirling number of the second kind $\{{m \atop k}\}$ counts.
