Incompatible babel-loader and babel-core versions

[email protected] is the Webpack integration used for Babel 7.x. Babel 7.x has moved all packages from a babel- prefix to the @babel npm scope.

The error is correct, you have incorrectly installed babel-core instead of @babel/core.

If you wish to install Babel 6.x, you can do

npm install --save-dev babel-loader@7 babel-core

but if you're starting a new project, Babel 7 makes much more sense, so you would ideally do

npm install --save-dev babel-loader @babel/core