That's fill up as in no more room to store anything else on the drive that only contains zero size files. Can you figure out how and why before you watch the video?
Simple: Every file created creates an entry in the file system (contains the name of the file, date created, date modified, who its parent directory is, what sectors the data resides in, etc.).
Applications usually have small files packed into archives to cut back on the file system overhead as well as improve access times. A directory full of lots of little files can actually occupy more space in terms of file system than actual data each file contains.
Edit: Terraria is an example of a game with a lot of file system overhead due to having over 9000 files and a lot of them <1 KiB:
Note the difference between "Size" (content data) versus "Size on disk" (content data + NTFS overhead). ~6 MiB of file metadata.