How can I get an intuitive explanation of the visualization of geodesics in a cone?

Let's use the rotational symmetry of the problem to set $alpha = 0$. The value of $r$ has to be continuous along your geodesic, which means that if $r > 0$ to begin with, then any point where $r to infty$ has to be regarded as the "end" of the geodesic. We're therefore restricted to $r < infty$, which means that over the range of $phi$ that is valid you must have $$ cos (omega phi) > 0. $$ This implies, since the inverse cosine function is multi-valued and $cos (pm pi/2) = 0$, that the allowable range of $phi$ is $$ -frac{pi}{2 omega} < phi < frac{pi}{2 omega} $$ But if $omega < frac12$, then the allowable range of $phi$ runs from an angle less than $- pi$ to an angle greater than $+pi$. There will therefore be a range of angles for which there are two possible values of $r$: one when $phi > 0$ and one when $phi < 0$. What's more, since $cos( - omega pi) = cos (+ omega pi)$ no matter what $omega$ is, the two $r$ values will be the same when $phi = pm pi$, and the curve will intersect itself there.

If you plug in a value for $phi$ outside of the above range, you will get a result where $-infty < r < 0$. But this curve can't be can't be continuously connected to the same geodesic as the "original" $ r>0$ piece we talked about above. Rather, it can be thought of as a "different" geodesic with a different set of initial conditions.