The author is rewriting their React code to utilize Pure Components, a type of component that simply presents data without manipulating it. They were initially using Smart Components, which are aware of their own state and can manipulate it, but found it challenging to test and present. The author simplified the code by moving each section into its own function and creating a Pure Component for rendering plans. This refactoring resulted in fewer lines of code in the Smart Component and allowed for reusability of the PlanMenu component. The use of Pure Components enables more readable and maintainable code, making it easier to reuse components throughout the application.