Quiet a short algebraic proof may go as follows (I cite my own manuscript). For a polynomial $f(x_1,\dots,x_{n+1})$ of degree at most $n$ we define a linear operator $$\Phi[f]=\text{Sym}\, \frac{f(x_1,\dots,x_{n+1})}{(x_1-x_2)(x_2-x_3)\dots (x_n-x_{n+1})},$$
where $\text{Sym}\, g(x_1,\dots,x_{n+1})=\sum g(x_{\pi_1},\dots,x_{\pi_{n+1}})$, the summation is over all $(n+1)!$ permutations of the variables. Note that $\Phi[f]$ is actually a polynomial of degree at most 0 (all factors in the denominator are cancelled, see Ilya's comment), in other words, it is a scalar, and the linear operator $\Phi$ is a linear functional.
Several observations about $\Phi$.
1) If $f$ is divisible by $y:=x_1+\dots+x_{n+1}$, then $\Phi[f]=y\Phi[f/y]=0$. (The same holds for any symmetric factor $y$.)
2) Assume that $f=(x_{k}-x_{k+1})h(x_1,\dots,x_k)g(x_{k+1},\dots,x_{n+1})$. Then $$\Phi[f]=\begin{cases}\binom{n+1}k\Phi_k[h]\Phi_{n+1-k}[g],\,\text{if}\deg h=k-1\,\text{and} \deg g=n-k\\
0,\,\text{otherwise}\end{cases}.$$
3) If $f$ is divisible by $(x_1+\dots+x_k)(x_{k}-x_{k+1})$, then $\Phi[f]=0$. Indeed, it suffices to consider the polynomials $f=(x_1+\dots+x_k)(x_{k+1}-x_k)h(x_1,\dots,x_k)g(x_{k+1},\dots,x_{n+1})$. For them the claim follows from 1) and 2).
Before proving your Euler numbers claim we prove that $c_k:=\Phi[x_k^n]=(-1)^{k-1}\binom{n}{k-1}$. Induction in $n$. Base is straightforward. Induction step. We have $c_0+\dots+c_n=\Phi(x_1^n+\dots+x_{n+1}^n)=0$, by observation in 1). Next,
$$
c_k-c_{k+1}=\Phi((x_k-x_{k+1})(x_k^{n-1}+\dots+x_{k+1}^{n-1})=
(-1)^{k-1}\binom{n+1}k
$$
by 2) and by the induction assumption. We obtain $n+1$ linear relations on $c_1,\dots,c_{n+1}$ which determine
them uniquely and also the numbers $c_i=(-1)^{i-1}\binom{n}{i-1}$ satisfy these relations. This finishes the induction step.
Now we want to calculate $\Phi[(x_1+\dots+x_{k+1})^n]$. By 3) we have
$$
\Phi[(x_1+\dots+x_{k+1})^n-(x_1+\dots+x_{k}+x_k)^n+x_k^n-x_{k+1}^n]=0,
$$
thus $$\Phi[(x_1+\dots+x_{k+1})^n-(x_1+\dots+x_{k}+x_k)^n]=(-1)^{k}\binom{n+1}k.$$
Next, $$\Phi[(x_1+\dots+x_{k-1}+2x_k)^n-(x_1+\dots+x_{k-2}+3x_{k-1})^n]=(-1)^{k-1}2^n\binom{n+1}{k-1}.$$
Proceeding in this way and summing up we get the standard formula for Eulerian number (1) on this MathWorld page.
