Find out where credentials go?

I found a video, where someone showed a phishing site, where the login credentials get saved and sent to the owner of the site. I'd like to know, how you know what the site is doing in the background, for example sending the stuff to an email or something like that. The video that I'm talking about (hopefully you can understand what I mean after that): https://www.youtube.com/watch?v=UtNYzv8gLbs (0:40 - 1:30) How do you find something like that? Thanks for your help in advance, Till