When rotation is not about an arbitrary axis but along a principle $x$, $y$ or $z$ axis, the products of inertia both are $0$, and so the angular momentum points along a principle axis. Why do they resolve to $0$? The products of inertia are:
$$- \sum_i m_i(x_iz_i)$$ $$- \sum_i m_i(y_iz_i)$$
How do these go to $0$ if the angular momentum is along a principle axis?
TL;DR: Whether the products of inertia are zero or not depends on the choice of coordinate system, not on whether the object rotates along a principal axis.
In more details:
The products of inertia are the off-diagonal elements $I_{xy}$, $I_{yz}$, $I_{zx}$, of the inertia tensor, cf. this & this Phys.SE posts.
The inertia tensor is real & symmetric, and hence diagonalizable in an orthonormal real basis.
The principal axes are the directions of the eigenvectors.
If a Cartesian coordinate system has all 3 axes along the 3 principal axes, then all 3 products of inertia $I_{xy}$, $I_{yz}$, $I_{zx}$, vanish.
If a Cartesian coordinate system has the $z$-axis along a principal axis, then $I_{yz}=0=I_{zx}$, but $I_{xy}$ is not necessarily zero. However, if furthermore $I_{xx}=I_{yy}$, then $I_{xy}=0$ as well.
Products of inertia are roughly the measure of how rotating around one axis can give angular momentum with respect to another axis. The principle coordinate system is the one for which the angular momenta are decoupled, in the sense that rotating around one axis yields only angular momentum around that axis. That is why products of inertia are all zero in principle coordinate system.
