How does soap affect membrane permeability? Which component of the membrane does it affect?
@mdperry answered well. If you need some diagrams to help with understanding, this article Should be helpful.

As you surmised, when you add detergent, it is lysing some cells of the beetroot that contain pigment.  That pigment then gets released into solution.  I've annotated the image below for the micelles that gets formed, except I spelled it wrong :/