Before that, you needed a video player installed on your computer that then either loaded in-browser or played videos after download. You often needed multiple players installed as none supported every video format.
Don't attribute user interest (which is what made YouTube grow) to the technical aspects of the site. That is almost never the case.
If you wanted to share videos online, good luck? You needed somewhere you could upload them and a lot of forums had file size limits, so sometimes you just spread/discovered them via p2p services. For the non-technical person, there were a few places you could submit videos to or browse 'funny internet videos', but most video sites were an individual or community's curated collection
Youtube bundled all of that up as "a video repository and search engine that provides a streaming player which is already supported by your browser", and (in the US anyway) that moment coincided with: (a) wide-spread broadband access and (b) digital cameras becoming accessible to consumers
So in that context, I think it might be a bit like saying "interest in reading increased after book printing technology made libraries comparatively inexpensive"