The browser.js file is used by Opera to correct site specific problems that would otherwise make a site unusable in Opera.
This file is updated by Opera Software ASA automatically from time to time. If you would like to make sure that you have the latest browser.js file update, simply go to Help -> Check for Updates. This will update the browser.js file in the background if an update is available. You will not see any visual confirmation.
To check the browser.js version that you are using, simply navigate to the official browser.js page at Opera.com.
Editing this file in any way will cause Opera to stop using it. Browser.js is signed by Opera with each update, thus making it very secure and safe to use.
If you are looking for a way to correct other site problems similar to the way that browser.js is used, please look into using UserJS patches.