It's done in order to root the path (making it an absolute path).
It ensures that the path is not relative but read from the root of the site.
This allows one to move a file around and not have to change the links to the different resources.
Using your example:
src="/assets/js/jquery.js"
If the referencing file is in /pages/admin/main.html (for example) using relative paths you would use:
src="../../assets/js/jquery.js"
Suppose you move the file to a child directory. No changes would be needed for with the original rooted path, but the relative one would need to change to:
This is to ensure the asset comes from the "root" of the web server.
e.g.
Host is www.example.com
URL becomes www.example.com/assets/js/jquery.js
I do this with project I want to ensure live on their own virtual host.
The issue really comes down to where those assets are being included. For example if the asset is being included from /help/pages/faq then the developer can be sure the path will work correctly when the site is hosted on a non changing host, e.g. example.com.
The issue of using relative paths, 'assets/js/jquery.js' is that if the assets are included from the /help/pages/faqs then the path becomes relative to that starting point, e.g. /help/pages/faqs/assets/js/jquery.js
This is a bit off topic, but if there is any chance that your application will ever be served behind a reverse proxy (eg. using apache2 or nginx) under a sub-path, you should try to avoid absolute paths.
For example, if you reference "/style.css" on https://example.com/, and you tried to hide it behind a reverse proxy at https://proxy.example.com/example/, your absolute reference would break. The browser would make the request to "https://proxy.example.com/style.css" when it should have requested "https://proxy.example.com/example/style.css".
Unintentional absolute paths from a leading forward slash are a nightmare for reverse proxies to deal with.