Without Babel you had to use the feature set of your oldest browser or use feature detections such as modernizr or write polyfills on your own.
In general, Babel is split in 2 ways to bring you the new goodies you want.
New syntax will be compiled to old EcmaScript 5 code e.g.:
New globals and functions are provided by babel-polyfill e.g.:
If you are using TypeScript, Babel is not necessary since you already get the new features with TypeScript.