How do I get/scrape videos/livestreams displayed on a website?

I googled for a while now and I am not sure if I can ask my question properly. So, now I am here trying to "understand" what I want. Maybe you guys can help me.

When I go on a website that has a video feed (Youtube, Twitch, etc.) my browser displays that Video. So can I somehow access that video feed? Grab the pictures, the whole video, maybe access the cache and recover that video (although I read that many sites down store the video on your pc anymore)?

I've tried to look in the debugging menu of chrome to see if you can see a link where the video is coming from or if there are any series of pictures or videos stored that are simply played but I've found nothing.

I am sure there are a lot of cool APIs out there for youtube and twitch and so on but I really want to gain more of an understanding of this thing rather than just copy and paste code or use a "black box" command.

Thanks in advance.