How does the internet archive work? How can it emulate websites like that?

It's not emulating anything, it just shows you a snapshot of the page.

Wget <website.com>
Save for later in a db with a timestamp.

Render to user on request.

There's a library that archives everything based on references in the dom.

It should be OSS, last time I checked anyway.

I found couble crawler projects under Internet Archive's Github account.

https://github.com/internetarchive/...
https://github.com/internetarchive/...

Brozzler renders the page using a Chromium browses. From a first look I think it saves all the requests and responses to database for replaying. There might be some processing before the web page ends up in Wayback Machine, like changing hostnames.