I can understand your confusion. Imho this review is unreadable if you try to learn the topic for the first time.
For particle physicists, a potential for a dynamical field (one which has a kinetic term like $\partial_\mu a \partial^\mu a$) is nothing more than additional terms in the Lagrangian density. They can be field dependent, or also be constant. (e.g. a mass term in that sense is also just a potential with quadratic dependence on the field.)
Due to the non-trivial vacuum structure of QCD, there is a term in the QCD Lagrangian density that causes the Strong CP problem (this is not axion-dependent and exists in QCD without axions): $$\mathcal{L}_{\bar\theta}=\bar{\theta}\frac{g_s^2}{32\pi^2}\left<G^{\mu \nu}\tilde{G}_{\mu \nu}\right>. $$
In a theory with the axion, this represents a constant potential term for the axion.
However, due to a chiral anomaly, there is similar term in the Lagrangian density that is actually axion dependent, the one you mention above (compare (22) in the review):
$$\mathcal{L}_a\supset \frac{\xi\,a}{f_a} \frac{g_s^2}{32\pi^2}\left<G^{\mu \nu}\tilde{G}_{\mu \nu}\right>.$$
You see that there is a particular value for $a$, $-\bar\theta f_a/\xi$ for which both terms add to zero. This will be the value that the axion will take in a dynamical theory, because it minimizes the value of the sum of both of these potential terms, i.e. the axion's effective potential.
You added a comment how the potential effectively looks, with a cosine like structure. To find this is another story, because one has to evaluate above expressions in the low energy limit of QCD. This is were you find that the effective low energy potential actual has $\bar\theta+\xi a/f_a$ as argument, which makes the relation between $\bar\theta$ and $a$ explicit.
My tip: Search for more in depth literature, Peccei's review only gives the broadest step, which are, as mentioned, very confusing when you read them there for the first time.
