ES6 classes are here to help you make your code cleaner and organize it better, compared to the function inside of function thing which is ES5.
If you plan on building a big project which will be worked on for a long time and a lot of features will be added, then you have to figure out a good structure for your app.
You could put everything in one file and be done with it but when you get back to it one day, it will be chaos and you'll have a lot of fun figuring out what you did 6 months or 2 years ago.
To use ES6, use a tool like Babel or something similar until all of the browsers have 100% ES6 support.
ES6 helps you a lot if you know how to use it. Before you begin using it, make sure to study it in details so you don't get stuck in the middle of your development cycle.
JavaScript wasn't built for a lot of the features it has today but people still use it and it helps them, so using classes in a non-class language doesn't matter.