By way of enrichment here is an alternate formulation using
cycle indices.
Recall that the OGF of the cycle index $Z(P_n)$ of the unlabeled set
operator $\mathfrak{P}_{=n}$ is given by
$$G(w) = \sum_{n\ge 0} Z(P_n) w^n =
\exp\left(\sum_{q\ge 1} (-1)^{q+1} a_q \frac{w^q}{q}\right).$$
Differentiating we obtain
$$G'(w) = \sum_{n\ge 0} (n+1) Z(P_{n+1}) w^n =
G(w) \left(\sum_{q\ge 1} (-1)^{q+1} a_q w^{q-1}\right).$$
Extracting coefficients we thus have
$$[w^n] G'(w) = (n+1) Z(P_{n+1}) =
\sum_{q=1}^{n+1} (-1)^{q+1} a_q Z(P_{n+1-q})$$
This is apparently due to Lovasz.
Substitute the cycle indices with the variables $X_1$ to $X_m$ to get
$$(n+1) Z(P_{n+1})(X_1+\cdots+X_m) \\=
\sum_{q=1}^{n+1} (-1)^{q+1} (X_1^q+\cdots+X_m^q)
Z(P_{n+1-q})(X_1+\cdots+X_m)$$
This yields
$$(n+1) e_{n+1}(X_1,\ldots,X_m) =
\sum_{q=1}^{n+1} (-1)^{q+1} p_q(X_1,\ldots,X_m)
e_{n+1-q}(X_1,\ldots,X_m).$$
Now a choice of variable names yields the result.
Remark. The identity for $G(w)$ follows from the EGF for the
labeled species for permutations where all cycles are marked with a
variable indicating length of the cycle.
This yields
$$\mathfrak{P}
\left(A_1 \mathfrak{C}_{=1}(\mathcal{W})
+ A_2 \mathfrak{C}_{=2}(\mathcal{W})
+ A_3 \mathfrak{C}_{=3}(\mathcal{W})
+ \cdots \right).$$
Translating to generating functions we obtain
$$G(w) = \exp\left(a_1 + a_2 \frac{w^2}{2}
+ a_3 \frac{w^3}{3} + \cdots\right).$$
The fact that $$Z(P_n) = \left.Z(S_n)\right|_{a_q := (-1)^{q+1} a_q}$$
then confirms the claim.
