You should look at the form of the advanced fundamental solution of D'Alembert equation, built up in geodesically convex open sets including the source localized at the event $y$ and the test point localized at the enent $x$ receiving the wave generating by the source. The construction, at least for analytic manifolds with analytic metrics, is obtained by summing a nice series originally discovered by Hadamard (and handled by Riesz actually; there is indeed a wonderful paper in French by Riesz about this fantastic construction nowadays relating the heat kernel theory with QFT in curve spacetime). Hadamard-Riesz' results have been extended to the smooth case by sevaral modern authors (see Guenther's and Friedlander's textbooks). The series, if the dimension gives rise to a fundamental solution containing a term which is completely supported on the light cone emanating from $y$. Therefore, referring to this term only, the solutions of D'Alembert equation emitted by $y$ propagates along null geodesics to reach $x$ from $y$. This basically is Huygens' principle.
If the dimension is even and the manifold is not flat or the dimension is odd, further terms appear added to the one localized on the light cone. The underlying "mathematical phenomenon" is more or less the same, in flat spacetime, when adding a mass to D'Alembert operator thus passing to the Klein-Gordon equation which does not obey Huygens' principle.
The relevant point is that this further term is now supported inside the future light cone emanating from $y$. In this case there is a contribution to wave solutions emitted by $y$ propagating along timelike geodesics from $y$ to $x$, and Huygens' principle fails.