“You may need an appropriate loader to handle this file type” with Webpack and Babel

You need to install the es2015 preset:

npm install babel-preset-es2015

and then configure babel-loader:

{
    test: /\.jsx?$/,
    loader: 'babel-loader',
    exclude: /node_modules/,
    query: {
        presets: ['es2015']
    }
}