Note that the degenerate state here only applies to the electrons, the helium nuclei can still be flying around and having fusion in normal ways. Once helium fusion begins in the core, it causes changes in the rest of the star. The text does a good job of describing this behavior, I like its analogy of the boiling water in the lidded pan. Core produces energy. Outer layers opaque, trap radiation.
Outer layers heat and expand.
Layer becomes transparent and cools. Gases fall back inwards, become denser and opaque again. Named for the first studied stars of these types. All low mass stars will become RR Lyrae or Cepheid variables stars during their life. So these star types are very common.