You can parametrize the three-dimensional manifold $M$ by the magnitude of either $z_1$ or $z_2$ and the two arguments of $z_1$ and $z_2$. Let's call the one of $z_1$ and $z_2$ whose magnitude we use for the parametrization $z$ and the other $z'$. You can visualize this parametrization using the volume enclosed by a torus, with the magnitude of $z$ corresponding to the distance from the central ring, the argument of $z$ corresponding to the angle with respect to the central ring (the poloidal angle) and the argument of $z'$ corresponding to the angle with respect to the torus axis (the toroidal angle). That is, with the parameters used in the Wikipedia article on the torus, the magnitude of $z$ is $r$, the argument of $z$ is $v$, and the argument of $z'$ is $u$.
To see what happens in the two interesting neighbourhoods around $z_1=0$ and $z_2=0$, we can use the appropriate torus where $z$ is the one of $z_1$ and $z_2$ that is close to zero. Dealing first with $z_1$ near $0$ (and thus $z=z_1$, $z'=z_2$), the orbits of the given action of $S^1$ are spirals that spiral around the central ring $k$ times before returning to the same point. We can pick any disk defined by some value of $u$ as the set of representatives of these orbits, and see that the space of these orbits has the manifold structure of an ordinary disk.
On the other hand, if $z_2$ is near $0$ (and thus $z=z_2$, $z'=z_1$), the orbits are again spirals that spiral around the central ring, but this time, they only go around the ring $1/k$ times before returning to the same value of $u$, and only return to the same point after going around the torus axis $k$ times. If we now pick any disk defined by some value of $u$, it contains $k$ representatives of each orbit. Thus, the space of these orbits has the same structure as the quotient of the disk with respect to the cyclic group of rotations generated by a rotation around the origin through $2\pi/k$. The conical singularity arising from this is described here for the case $k=2$. Note that it is only a singularity in the differentiable structure, not in the topological structure, since the cone is topologically a two-dimensional manifold at the origin.