To start, your equation for the Stein operator (equation 1 of the paper) is the same as equation 3.4 of the Wikipedia article. If we write $q$ for your $p$ then
$$\nabla_x\log p(x)^\top$$
is Wikipedia's $q'(x)/q(x)$. Writing $f$ for your $\phi$,
$${\cal A}_p\phi(x)\equiv {\cal A}_pf(x)=f(x)q'(x)/q(x)+f'(x)$$
as Wikipedia has it.
Now, so what? Well, suppose we actually sample $X$ from a distribution $P$ but we want to approximate it by a simpler distribution $Q$ (using Wikipedia's notation). We want to choose (learn) $Q$, but (for some reason, perhaps extra coolness points) we don't want to restrict to a parametric family $Q_\theta$
Stein's Lemma says
$$E[{\cal A}_pf(X)]=0$$
for all (sufficiently nice) $\phi$ when $Q=P$. That sort of looks like the condition that the derivative in the 'direction' $f$ is zero when the distance between $Q$ and $P$ is minimised. So we might get the idea of using $E[{\cal A}_pf(X)]$ as a sort of 'gradient' of some distance $d(P,Q)$ in the 'direction' $f$ and doing something like coordinate descent or gradient descent by updating $$q(x)\mapsto q(x)-\epsilon \times E[{\cal A}_pf(X)]\times f(x)$$
What the paper does is show that the handwaving can be made precise, so you do get gradient descent for the Kullback-Leibler divergence. It also (crucially) shows that the properties holding for all (sufficiently nice) $f$ can be replaced by them holding for $f$ generated by some useful set of transformations of $x$ (a reproducing kernel Hilbert space). This means you get an algorithm that can actually be implemented
