(react-router-dom version: 4.1.1)

I have working routes set up, but I'm a bit confused about why the <Switch> was necessary:

index.js

import React from 'react';
import { HashRouter, Route, Switch } from 'react-router-dom';

import App from './components/App.jsx';
import FridgePage from './components/FridgePage.jsx';

ReactDOM.render(
    <HashRouter>
        <Switch>
            <Route exact path="/" component={App} />
            <Route path="/fridge" component={FridgePage} />
        </Switch>
    </HashRouter>,
    document.getElementById('root')
)

App.jsx

import Header from './Header.jsx';
import {Link} from 'react-router-dom';

export default class App extends React.Component {
    render() {
        console.log(this.props);
        return (
            <div>
                <h1>Herbnew</h1>
                <Link to="fridge">Fridge</Link>
                {this.props.children}
            </div>
        );
    }
}

FridgePage.jsx

import React from 'react';

export default class FridgePage extends React.Component {
    render() {
        return (
            <div>
                <h1>Fridge</h1>
                You finally found the fridge!
            </div>
        );
    }
}

I used to have a div wrapping the routes instead of a Switch. In that case, I see the App rendered and try to click the Fridge link, but nothing happens (the FridgePage isn't rendered), and no error is output into the console.

As I understand it, the only thing the Switch does is exclusively render the first route it matches, and the common problem as a result of omitting it is rendering both pages at once. If my "/" route is exact, then even without the Switch, the Fridge should be the only route that matches, right? Why does it not render at all?