Polylogarithms of negative integer order
The Wiki page presenting polylogarithms states identities with Stirling numbers of the second kind, one of them being \begin{align*} \mathrm{Li_{-n}}(z)=\sum_{k=0}^{n}k! \begin{Bmatrix}n+1\\k+1\end{Bmatrix} \left(\frac{z}{1-z}\right)^{k+1}\qquad\qquad n\geq 0\tag{1} \end{align*} and an identity with Eulerian numbers \begin{align*} \mathrm{Li_{-n}}(z)=\frac{1}{(1-z)^{n+1}}\sum_{k=0}^{n-1}k! \left\langle n \atop k \right\rangle z^{n-k}\qquad\qquad n\geq 1\tag{2} \end{align*} According to OPs question I'll try to provide a plausible reasoning how these formulae could be derived.
First part: Stirling numbers of the Second kind
In order to derive the identity (1) a crucial observation is the recurrence relation of the polylogarithm regarding the differential operator $\mathrm{D}=\frac{d}{dz}$. We can write for $n\geq 1$ \begin{align*} \mathrm{Li_{-n}}(z)=(z\mathrm{D})\mathrm{Li_{-n+1}}(z) \end{align*}
From this relation we obtain by successive application of the operator $z\mathrm{D}$ as already shown in OPs question
\begin{align*} \mathrm{Li_{-n}}(z)=(z\mathrm{D})\mathrm{Li_{-n+1}}(z)=\ldots =(z\mathrm{D})^n\mathrm{Li_{0}}(z)=(z\mathrm{D})^n\frac{z}{1-z}\tag{3} \end{align*}
The connection with Stirling numbers of the second kind is given via the operator $z\mathrm{D}$, since the following well known operator relation is valid
\begin{align*} (z\mathrm{D})^n=\sum_{k=1}^n\begin{Bmatrix}n\\k\end{Bmatrix}z^k\mathrm{D}^k\qquad\qquad n\geq 1\tag{4} \end{align*} Information about this relation can be found e.g. in Stirling numbers of the second kind by R. Milson.
Applying the relation (4) to $\frac{z}{1-z}$ we obtain for $n\geq 1$ \begin{align*} (z\mathrm{D})^n\frac{z}{1-z}&=\sum_{k=1}^n\begin{Bmatrix}n\\k\end{Bmatrix}z^k\mathrm{D}^k\frac{z}{1-z}\\ &=\sum_{k=1}^n\begin{Bmatrix}n\\k\end{Bmatrix}z^k\mathrm{D}^k\sum_{j=1}^{\infty}z^{j}\\ &=\sum_{k=1}^n\begin{Bmatrix}n\\k\end{Bmatrix}z^k\sum_{j=k}^{\infty}k!\binom{j}{k}z^{j-k}\tag{5}\\ &=\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}z^k\sum_{j=0}^{\infty}\binom{j+k}{k}z^{j}\\ &=\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}z^k\sum_{j=0}^{\infty}\binom{-(k+1)}{j}(-z)^j\tag{6}\\ &=\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\frac{z^k}{(1-z)^{k+1}}\tag{7} \end{align*}
Comment:
In (5) we differentiate $k$ times and use $j(j-1)\cdots(j-k+1)=k!\binom{j}{k}$
In (6) we use $\binom{-n}{k}=\binom{n+k-1}{k}(-1)^k$
In (7) we apply the binomial series
Combining the polylogarithm representation (3) with (7) we obtain
\begin{align*} \mathrm{Li_{-n}}(z)=\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\frac{z^k}{(1-z)^{k+1}} \qquad\qquad n\geq 1\tag{8} \end{align*}
We have found a relationship beween the polylogarithm and the Stirling numbers of the second kind. Note, this is not the representation (1) stated in the Wiki page, but we will show equality. In order to do so we recall the recurrence relation of Stirling numbers of the second kind \begin{align*} \begin{Bmatrix}n+1\\k\end{Bmatrix}=k\begin{Bmatrix}n\\k\end{Bmatrix}+\begin{Bmatrix}n\\k-1\end{Bmatrix} \qquad\qquad n\geq 0, k\geq 1 \end{align*} and claim
The following holds true \begin{align*} \sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\frac{z^k}{(1-z)^{k+1}}= \sum_{k=0}^{n}k!\begin{Bmatrix}n+1\\k+1\end{Bmatrix}\left(\frac{z}{1-z}\right)^{k+1} \qquad\qquad n\geq 1 \end{align*}
We start with the RHS and get \begin{align*} \sum_{k=0}^nk!&\begin{Bmatrix}n+1\\k+1\end{Bmatrix}\left(\frac{z}{1-z}\right)^{k+1}\\ &=\sum_{k=0}^nk!\left((k+1)\begin{Bmatrix}n\\k+1\end{Bmatrix}+ \begin{Bmatrix}n\\k\end{Bmatrix}\right)\left(\frac{z}{1-z}\right)^{k+1}\tag{9}\\ &=\sum_{k=0}^n(k+1)!\begin{Bmatrix}n\\k+1\end{Bmatrix}\left(\frac{z}{1-z}\right)^{k+1} +\sum_{k=0}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\left(\frac{z}{1-z}\right)^{k+1}\\ &=\sum_{k=1}^{n+1}k!\begin{Bmatrix}n\\k\end{Bmatrix}\left(\frac{z}{1-z}\right)^{k} +\sum_{k=0}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\left(\frac{z}{1-z}\right)^{k+1}\tag{10}\\ &=\sum_{k=1}^{n}k!\begin{Bmatrix}n\\k\end{Bmatrix}\left[\left(\frac{z}{1-z}\right)^{k} +\left(\frac{z}{1-z}\right)^{k+1}\right]\\ &=\sum_{k=1}^{n}k!\begin{Bmatrix}n\\k\end{Bmatrix}\frac{z^k}{(1-z)^{k+1}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box \end{align*}
Comment:
In (9) we apply the recurrence relation to $\begin{Bmatrix}n+1\\k+1\end{Bmatrix}$
In (10) we shift the index in the left sum by one and note that in the left sum the upper limit $k=n+1$ gives a factor $\begin{Bmatrix}n\\n+1\end{Bmatrix}=0$ and in the right sum the lower limit $k=0$ gives a factor $\begin{Bmatrix}n\\0\end{Bmatrix}=0$. This is respected in the following line.
Conclusion: The differential operator $z\mathrm{D}$ provides a convenient connection between the polylogarithm $\mathrm{Li_{-n}}(z)$ of negative integer order and the Stirling numbers of the second kind.
The Eulerian numbers are strongly connected to Stirling numbers of the second kind. So, it's plausible that there is also a nice representation of the polylogarithm in terms of Eulerian numbers.
Second step: Eulerian numbers
In order to show the relation (2) stated in the Wiki page, we consider the Eulerian polynomials $A_n(z)$ defined as \begin{align*} A_n(z)=\sum_{k=0}^{n-1}\left\langle n \atop k \right\rangle z^k\qquad\qquad\qquad n\geq 1 \end{align*} These polynomials are linked with Stirling number of the second kind by a Theorem by Frobenius. See e.g. the Identity section in Eulerian polynomials. The theorem states \begin{align*} A_n(z)=\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}(z-1)^{n-k}\qquad\qquad n\geq 1 \end{align*}
With this theorem it's easy to derive the relation (2). We claim
The following holds true \begin{align*} \frac{1}{(1-z)^{n+1}}\sum_{k=0}^{n-1}\left\langle n \atop k \right\rangle z^{n-k}= \sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\frac{z^k}{(1-z)^{k+1}} \qquad\qquad n\geq 1 \end{align*}
This is valid since we obtain for $n\geq 1$ \begin{align*} \frac{1}{(1-z)^{n+1}}\sum_{k=0}^{n-1}\left\langle n \atop k \right\rangle z^{n-k} &=\frac{z^n}{(1-z)^{n+1}}\sum_{k=0}^{n-1}\left\langle n \atop k \right\rangle \left(\frac{1}{z}\right)^k\\ &=\frac{z^n}{(1-z)^{n+1}}\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\left(\frac{1}{z}-1\right)^{n-k}\tag{11}\\ &=\frac{z^n}{(1-z)^{n+1}}\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\left(\frac{1-z}{z}\right)^{n-k}\\ &=\sum_{k=1}^nk!\begin{Bmatrix}n\\k\end{Bmatrix}\frac{z^k}{(1-z)^{k+1}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box \end{align*}
Comment:
In (11) we apply Frobenius Theorem.
Note also that we used here as a matter of convenience the derived identity (8) and not the identity stated in the Wiki page.
Suppose we seek to show that $$\mathrm{Li}_{-n}(w) = \frac{1}{(1-w)^{n+1}} \sum_{k=0}^{n} \left\langle {n\atop k}\right\rangle w^{n-k}.$$
We are done if we can prove that ($\left\langle {n\atop n}\right\rangle$ is zero) $$[w^m] \frac{1}{(1-w)^{n+1}} \sum_{k=0}^{n-1} \left\langle {n\atop k}\right\rangle w^{n-k} = m^n.$$
Extracting coefficients we get $$\sum_{k=0}^{n-1} \left\langle {n\atop k}\right\rangle {m+k-n + n\choose n} = \sum_{k=0}^{n-1} \left\langle {n\atop k}\right\rangle {m+k\choose n}.$$
Start from the bivariate generating function of the Eulerian numbers, which seems like a reasonable starting point and which is $$\frac{u-1}{u-\exp((u-1)z)}.$$
Now introduce $${m+k\choose n} = \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m+k}}{v^{n+1}} \; dv.$$
This gives for the sum $$\frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m}}{v^{n+1}} \sum_{k=0}^{n-1} \left\langle {n\atop k}\right\rangle (1+v)^k \; dv.$$
Substitute the generating function of the Eulerian numbers into this to get $$\frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m}}{v^{n+1}} n! [z^n] \frac{v}{1+v-\exp(vz)} \; dv \\ = n! [z^n] \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m}}{v^{n+1}} \frac{1}{1-(\exp(vz)-1)/v} \; dv$$
Observe that $(\exp(vz)-1)/v$ expanded about zero with respect to $z$ starts at $z.$ Since we are extracting the $[z^n]$ coefficient we may write $$n! [z^n] \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m}}{v^{n+1}} \sum_{p=0}^n \frac{(\exp(vz)-1)^p}{v^p} \; dv.$$
We do the coefficient extraction with the integral $$\frac{n!}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} (\exp(vz)-1)^p \; dz$$
Put $vz = w$ so that $dz = dw/v$ to get $$\frac{n!}{2\pi i} \int_{|w|=\epsilon} \frac{1}{(w/v)^{n+1}} (\exp(w)-1)^p \; \frac{1}{v} dw \\ = \frac{n! \times v^n \times p!}{2\pi i} \int_{|w|=\epsilon} \frac{1}{w^{n+1}} \frac{(\exp(w)-1)^p}{p!} \; dw = v^n \times p! \times {n\brace p}.$$
Returning to the main integral we thus obtain $$\frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m}}{v^{n+1}} \sum_{p=0}^n v^{n-p} \times p! \times {n\brace p} \; dv \\ = \frac{1}{2\pi i} \int_{|v|=\epsilon} \frac{(1+v)^{m}}{v^{p+1}} \sum_{p=0}^n p! \times {n\brace p} \; dv = \sum_{p=0}^n p! \times {n\brace p} \times {m\choose p}.$$
To conclude observe that $$m^n = \frac{n!}{2\pi i} \int_{|z|=\epsilon} \frac{\exp(mz)}{z^{n+1}} \; dz = \frac{n!}{2\pi i} \int_{|z|=\epsilon} \frac{\exp(z)^m}{z^{n+1}} \; dz \\ = \frac{n!}{2\pi i} \int_{|z|=\epsilon} \frac{(1+(\exp(z)-1))^m}{z^{n+1}} \; dz \\ = \frac{n!}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \sum_{p=0}^m {m\choose p} (\exp(z)-1)^p \; dz \\ = \frac{n!}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \sum_{p=0}^m p! \times {m\choose p} \frac{(\exp(z)-1)^p}{p!} \; dz \\ = \sum_{p=0}^m p! \times {m\choose p} \times {n\brace p}.$$
Since the effective upper limit is always $\min(n,m)$ which is symmetric we may use either $m$ or $n$ as the upper limit and we are done.
Alternatively when $n \lt m$ we may change the upper limit from $m$ to $n$ because the Stirling number is zero for $n\lt p \le m.$ If $n \gt m$ we may also set the upper limit to $n$ because the binomial coefficient is zero for $n\ge p \gt m.$
Remark. This last sum may also be evaluated combinatorially. Count all functions from $[1,n]$ to $[1,m].$ To do this first fix the number $p$ of different values that will occur, next choose a set partition of $n$ into $p$ sets the constituents of each of which will be mapped to the same value, and finally choose the $p$ values from $m$ to use and some ordering thereof, giving the result
$$m^n = \sum_{p=0}^n p! \times {n\brace p} \times {m\choose p}.$$