How do people use video game footage in their youtube videos?

A lot of people have a card or some piece of hardware that they insert into the guts of the console that records their screens and it either connects to a storage device so they can export them to their computer or they connect the thing to the internet and set it up so the video clips are emailed to the person or something along those lines. Now with the Xbox One, people can do it with the new app that allows you to record your screen, I even saw it record my screen for no reason and I didn't even tell it to record the screen: it was a dope play in NBA 2k14 so apparently there's something in the game that tells the Xbox One to record that clip which probably means there's a constant recording of your screen (I believe there's an option that allows you to turn that off though).

People either record their consoles with capture cards or download footage off youtube or any other site. You can use this program to download a video off youtube. http://www.any-video-converter.com/products/for_video_free/

Use bandicam to record gameplay.
Then just post your video with your gameplay onto youtube.

From a legal aspect? Well think of it as advertising, When you put a gaming video on youtube you are in effect advertising the game making more people want to play, companies can't complain! Although some do, there are a lot of companies that allow moetization and legal rights to be able to upload the clips and make money. Notice how pewdiepie plays a lot of indie games? This is because the publishers allow you to upload these clips, there are a couple of companies like Nintendo who sometimes don't allow monetization, same with 343 and bungie some people say they had there video removed.

The legal part to upload gaming is to add your own bit, I.E commentary! Without commentary it is a bit risky… But it depends, pewdiepie is in a network and his network can help him with copyright, networks have certain licenses they pay for so people can monetize videos, obviously they get it back in monetized views.

So in effect it wouldn't be ideal uploading someone elses clip or having no commentary or something you put in yourself because your not contributing (in a way). But having videos on there's advertising for companies.

I record videos, i have 100 present that includes Minecraft, GTA, Terraria, Halo, and a few indie games i have played, i'm in a network and have never been flagged for copyright.

There's a small list here that may help you understand a little better of what i'm saying:

Hope it helps!