I wouldn't take the term "fractal" too seriously (or at least too visually).
Basically, they prove that the generating function of $p(n)$ (which happens to be a modular form) has nice congruence properties modulo powers of $p$ when hit with the $U_{p^2}$ operator repeatedly. This latter operator has the effect $\sum a_nq^n\mapsto \sum a_{p^2n}q^n$ on the generating series, and thus can be thought of loosely as "$p$-adically zooming in" on the expansion. Hence the $p$-adically fractal turn of phrase.