When someone says they are 'changing their name servers', they normally mean that they are changing their name server address. A name server is actually a computer server that tells web browsers (like Chrome, Safari and Firefox) the location of your content, so it can load the pages for people visiting your site.
A useful way to think about it is to think of name servers as a phone number of where the content is located - and that browsers need to know that 'phone number' so they can 'dial' for the content (at the host) and load the page.
This all happens in a split second of course.
When you update your name servers to Ezoic, the Ezoic name servers delivers the address of the content, and Ezoic serves the files to the browser.
This is how Ezoic can deliver the new page layouts without changing your original content.
You can change your nameservers wherever you keep your domain registered (GoDaddy, Register.com, Enom, Tucows, etc)
Here is a video that helps explain all this.