The null hypothesis is that the differences are symmetrically distributed around zero. A small p-value means that the null hypothesis is rejected, i.e., that there is evidence against this. This may be because the center is not zero, or because the distribution is asymmetric. There's nothing stopping you from applying the test to asymmetrically distributed data, however you may not learn what you want to learn, because a rejection may be caused by the asymmetry, whereas you may be interested in whether the center is zero.
Note that in case of asymmetric distributions the definition of the "center" is ambiguous, it could refer to the median, the mean, or the mode, which will for asymmetric data normally differ from each other. So if what you are really interested in is whether the differences are centered around zero, in case of asymmetric data your test problem isn't well defined. However, a non-rejection of the $H_0$ by Wilcoxon's Signed Rank Test could still be interpreted as not giving any evidence that the differences are located anywhere else than around zero (in the sense that the rank sums on both sides are about the same).