What is cancel culture?
The widespread online shaming, ostracizing, villifying, condemnation, etc. of an individual in a community by other members of that community.
Why is it prominent?
It's prominent because it gets a lot of online coverage and it plays on our fears and emotions
Tied to (branch of philosophy)
Broadly tied to ethics, legal, social/political philosophy, and justice in general. In a more specific sense it's tied to a theory of retributive justice which is defended by ancient thinkers like Moses ("eye for an eye" etc.)
This might be a good link to do more research with: https://plato.stanford.edu/entries/justice-retributive/