I just tested it out.
It's pretty much the same thing as the Developer Tools you find in modern browsers, except that you are using the 3DS in order to analyze the rendering of the web pages.
You download DeveloperTools.zip directly from the 3DS by using the given URL, then extract it on your computer. The zip contains an index.html and resources folder with .js and images.
When you open index.html on your computer's web browser, you get a Javascript prompt asking for the IP address of the 3DS which was given earlier when you downloaded DeveloperTools.zip.
When you have done that, you are taken to a page showing all of the currently open tabs on the 3DS browser. When you enable the Developer Tools on the 3DS, all your previously open tabs and cookies will be deleted, showing only the main homepage of the 3DS browser. Selecting a tab on the computer front-end will take you to a page that acts as the developer tools on a modern browser.
Then it's pretty much the same as using a web browser's developer tools, with the computer working as the tools and the 3DS as the webpage renderer.
It's an interesting detail that Nintendo added to help web developers working for 3DS compatibility.
(Edit: Details added for clarity of how it works.)
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