Check it yourself
You do not have to take this on faith. Open your browser's developer tools, switch to the network tab, and convert a file. You will see the page and its scripts load, and you will see requests for ads and analytics. You will not see your file go anywhere — no request carries it, and no request is remotely the right size.
What is actually happening
Three pieces of the modern browser make this possible. The first is that a file you choose is handed to the page as a reference to something on disk, not as a lump of data. The page can ask for bytes 40,000 to 48,000 and get exactly those, which is why a two gigabyte file opens instantly and uses very little memory.
The second is a container parser written in JavaScript that reads the structure of an MP4 or an MKV — where the tracks are, where each compressed chunk sits — by asking for those small ranges rather than by reading the whole file.
The third is WebCodecs, a browser feature that exposes the same audio decoder the browser already uses to play media. Decoding runs at native speed on hardware built for it, which is why this is fast rather than a novelty.
Encoding to MP3 is the one piece browsers do not provide, so the page loads a compiled MP3 encoder, about 300 kilobytes, the first time you choose MP3. It runs in your browser like everything else.
Why other sites upload
Mostly because they were built before these features existed, and rebuilding a working product is expensive. A server also lets a site queue work, apply limits by plan, and keep the conversion out of reach — all things that make sense as a business and none of which make the result better for you. The visible consequence is the upload wait and the size cap.
What this does not protect
Your file staying local is a real and checkable property. It is not the same as the site being anonymous: this page serves ads and uses analytics, both of which see that you visited, as they would on any site. The precise claim is that the media never leaves your device, and that is the claim worth checking.