If the student streams on Twitch/YouTube etc they are probably already familiar with how to bring their footage over to their computer. 

If the student already has a preferred video editing software that is great! If not, I recommend starting with DaVinci Resolve because it is very high-end but offers a free version that has most of the functionality that students will need.

Finally, video files start to get large rather quickly and they will accumulate as students do more video work. I do recommend getting an external hard drive so that they are not filling up a family computer hard drive. This is not necessary, but it will help avoid a surprise situation down the road. Look for USB 3 and a size of 1TB or more. For students recording gameplay on a PC (or Mac) I recommend having something like OBS Studio set up to record straight from that PC on to an external storage device.

In addition to the Outschool classroom, this class uses:
OBS Studio
DaVinci Resolve

Two pieces of software that I list below are entirely optional. DaVinci Resolve is a video editing software available from BlackMagicDesign, regarded as a professional standard for decades. They offer a free version of their highly regarded software and I recommend it regularly to students who do not have their own video editing software. OBS Studio is a free screen recording software that makes game capture really simple. It has a very active development community and since it is open source, has remained free.

The majority of students want to make montages of themselves playing games like Fortnite or Destiny. These are shooter games that by their very nature do depict shooting, guns, grenades, explosions, and shooting at other people in-game. I personally will not be showing any footage from any game rated higher than T for Teen (I mostly play Destiny 2) but I cannot fully control what kinds of other games or content other students are allowed to play or what they discuss.