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An EPUB file is a book. If you have one on your device, you do not need an app to read it, and you do not need to upload it anywhere. This page opens EPUB files directly in the browser, remembers where you stopped, and works with the network off once it has loaded.
Your place in each book is saved on the device. Close the page, come back later, and it opens where you stopped. The highlights can be exported as plain text when you want to keep them somewhere else.
The two formats look similar and behave very differently. An EPUB file is a single archive of text, images and styles, which means the reader decides how to lay it out. Change the font size and the text reflows. Change the theme and the whole book turns dark. Search works. The page knows where you are.
A PDF is a picture of a page. It was designed to look the same everywhere, and it succeeds at that, which is exactly the problem when you are reading on a phone: the text is whatever size the designer chose, usually too small, and zooming in means losing your place. If you bought a book as EPUB, a PDF copy of the same book is usually worse to read on a screen.
Nowhere. The file is unpacked and read inside the browser using JavaScript, and it is never sent to a server. That matters for two reasons: it works offline the moment the page has loaded, and nobody else can read what you are reading. The same is true of your reading position and your highlights, which stay in the browser on that device.
What is an EPUB file? explains how the format works, what it is made of, and why publishers use it instead of PDF for books meant to be read on a screen.