When you copy something, most apps put several versions of it on the clipboard at once - a web page copy usually carries both styled text/html and a fallback text/plain, and the app you paste into picks whichever it prefers. This inspector reveals all of them so you can see exactly what will paste and why formatting sometimes carries over when you did not expect it.
Everything runs entirely in your browser. Your clipboard is read locally with the paste event or the browser's Clipboard API and is never uploaded, logged, or sent anywhere.