There are probably about 20 GBs of videos that I want to keep on my own Google Drive that I found publicly available on someone's Google Drive. I tried adding the files via the option Add to my Drive but that only added a shortcut on my drive. When I check the size of my drive, It says that it's about 278 MBs including 44 MBs of GMail. How can I keep a full-fledged copy of all the files without painstakingly downloading them and then uploading them manually?
On the list of files on Google Drive, over the file name do Right clic > Make a copy
After copying the file to your Drive, you can change the name of the file. Right click (or two-finger tap on a trackpad) the name of the file and select "Rename" and then you can change it to your preferred name.