In the context of Petr-Douglas-Neumann (PDN), "centroid" means "vertex-centroid" (ie, the average of the vertices). To see why the preservation of the vertex-centroid is "obvious" (and the least-interesting thing about PDN), let's consider a $n$-gon $p_0p_1p_2\cdots p_{n-1}$ with vertices $p_k$ in the complex plane.
Each step of the PDN process effectively applies a common operation to rotate-and-scale each side $p_kp_{k+1}$ about vertex $p_k$; we then shift focus to consider the polygon whose vertices are the images of the $p_{k+1}$. Each such image is given by
$$p_k + t (p_{k+1}-p_k) = (1-t) p_{k}+ tp_{k+1} \tag1$$
for some complex $t := s e^{i\theta}$, where $s$ is the (real) scale factor and $\theta$ the (real) rotation angle. (The exact values of, or relations between, $s$ and $\theta$ don't matter for this discussion.) We can represent this effect across the entire polygon by using matrices. Defining
$$P := \begin{bmatrix}p_0 & p_1 & p_2 & \cdots & p_{n-1}\end{bmatrix}
\qquad\text{and}\qquad T_t :=
\begin{bmatrix}
1-t & 0 & 0 & \cdots & t \\
t & 1-t & 0 & \cdots & 0 \\
0 & t & 1-t & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1-t
\end{bmatrix} \tag2$$
we have that the entries of $PT_t$ are the vertices of the resulting polygon; likewise, the entries of $PT_t T_u$, $PT_t T_u T_v$, etc, are the vertices of polygons from further iterations of the process.
Now, the vertex-centroid of a polygon is the average of its vertices; ie,
$$k := \frac1n(p_0+p_1+p_2+\cdots+p_{n-1}) = \frac1n P \,\mathbf{1}_n \tag3$$
where $\mathbf{1}_n$ is the "all-$1$s" column vector. But note: $T_t\,\mathbf{1}_n$ is just $\mathbf{1}_n$ again, so the vertex-centroid of the polygon obtained from applying the $T_t$ operation to $P$ is
$$\frac1n(P\,T_t)\,\mathbf{1}_n = \frac1n P\,(T_t\,\mathbf{1}_n) = \frac1n P\,\mathbf{1}_n = k \tag4$$
That is to say: the vertex-centroid of the original polygon $P$ is preserved under the $T_t$ operation; and clearly, it's preserved under every $T$-operation applied thereafter. The thing never moves! $\square$
Of course, centroid-preservation is a worthwhile property to mention, but its significance pales in comparison to the fact that the PDN process ultimately results in a (standard, convex) regular $n$-gon from any arbitrary $n$-gon, thus generalizing Napoleon's theorem for triangles. Also interesting: (1) the order of the steps doesn't matter; and (2) if we replace a step with one based on that "final" $n$-gon, then the new "final" figure corresponds to the step we replaced! (I'm being a bit loosey-goosey with the description here. In any case, showing these aspects requires getting into the details of the various $s$- and $\theta$-values, so I'll leave that as an exercise to the reader.)
