How to fetch the new data in response to React Router change with Redux?

Now, my question is, Is this really necessary or just a way to keep the architecture react-agnostic? Can I just fetch the information I need from react componentDidMount() and componentDidUpdate() instead?

You can totally do that in componentDidMount() and componentWillReceiveProps(nextProps).
This is what we do in real-world example in Redux:

function loadData(props) {
  const { fullName } = props;
  props.loadRepo(fullName, ['description']);
  props.loadStargazers(fullName);
}

class RepoPage extends Component {
  constructor(props) {
    super(props);
    this.renderUser = this.renderUser.bind(this);
    this.handleLoadMoreClick = this.handleLoadMoreClick.bind(this);
  }

  componentWillMount() {
    loadData(this.props);
  }

  componentWillReceiveProps(nextProps) {
    if (nextProps.fullName !== this.props.fullName) {
      loadData(nextProps);
    }

  /* ... */

}

You can get more sophisticated with Rx, but it’s not necessary at all.