The Weyl "algebra" $\mathbb{Z}[t, \partial_t]$ acts faithfully on the polynomial ring $\mathbb{Z}[t]$ in the obvious way. The latter is graded by degree, $t$ raises degree by $1$, $\partial_t$ lowers degree by $1$, and $t \partial_t$ preserves degree: in fact $(t \partial_t) t^m = m t^m.$
Thus $(t \partial_t)^n t^m = m^n t^m.$
A basis for the space of degree-preserving elements of the Weyl algebra is given by $t^k \partial_t^k, k \in \mathbb{Z}_{\ge 0}$, and $(t^k \partial_t^k) t^m = m(m-1)...(m-k+1) t^m = (m)_k t^m$
where $(m)_k$ denotes the falling factorial. Thus in order to find coefficients $a_{n,k}$ such that $(t \partial_t)^n = \sum_k a_{n,k} t^k \partial_t^k$
it is necessary and sufficient to find coefficients $a_{n,k}$ such that $m^n = \sum_k a_{n,k} (m)_k$
for all $m$. Since the polynomials $(m)_k$ form a basis of the space of polynomials, the coefficients $a_{n,k}$ exist uniquely, and in fact this is one way to define the Stirling numbers of the second kind.
The combinatorial interpretation is as follows: $m^n$ counts the number of functions $[n] \to [m]$, where $[n] = \{ 1, 2, ... n \}$. The above identity groups these functions together according to the size of their range: there are $(m)_k$ possible ranges of size $k$ and $a_{n,k}$ functions $[n] \to [m]$ having range a fixed subset of $[m]$ of size $k$. (This is the same thing as an equivalence relation on $[n]$ with $k$ equivalence classes by taking preimages.)