Managing state in Vue applications can be challenging due to the various approaches developers might take, but Vuex offers a structured solution for consistent state management across an app. In a tutorial by Mike Hartington, the process of implementing Vuex in a basic CRUD "TODO" application is discussed, highlighting the setup of a global store that manages the state. The tutorial explains how Vuex utilizes concepts like stores, mutations, and actions to handle state mutations and asynchronous tasks. Mutations allow for direct state changes, while actions facilitate asynchronous operations and can trigger multiple mutations. The use of TypeScript is also demonstrated to ensure type safety and enhanced developer experience through features like auto-completion. By organizing the application with Vuex, developers can achieve predictability and maintainability, making it easier to handle state changes and update the architecture as needed.