What is the light line or light cone and how is related with total internal reflection and snell law?

I’m have been reading about light propagation in periodic dielectrics. And in one part they introduce the light line. They mention that whenever the parallel and vertical components of the wave vector are bigger than the ones from the dispersion relation then there are not guided mode. In the contrary case they are guided modes.

Why those statements are true?

I attach a photo from a presentation where they made the same remarks.