Create a Redux store with createStore.

import { createStore } from 'redux'
import todoApp from './reducers'
let store = createStore(todoApp, { inistialStateVariable: "derp"})

Use connect to connect component to Redux store and pull props from store to component.

import { connect } from 'react-redux'

const VisibleTodoList = connect(
  mapStateToProps,
  mapDispatchToProps
)(TodoList)

export default VisibleTodoList

Define actions that allow your components to send messages to the Redux store.

/*
 * action types
 */

export const ADD_TODO = 'ADD_TODO'

export function addTodo(text) {
  return { type: ADD_TODO, text }
}

Handle these messages and create a new state for the store in reducer functions.

function todoApp(state = initialState, action) {
  switch (action.type) {
    case SET_VISIBILITY_FILTER:
      return Object.assign({}, state, {
        visibilityFilter: action.filter
      })
    default:
      return state
  }
}

Remarks

While React's component driven architecture is fantastic for breaking down the application into modular, encapsulated little pieces, it introduces some challenges for managing the state of the application as a whole. The time to use Redux is when you need to display the same data across more than one component or page (aka route). At that point you can no longer store the data in variables local to one component or the other, and sending messages between components quickly becomes a mess. With Redux your components are all subscribing to the same shared data in the store and thus the state can be easily reflected consistently across the entire application.