I don't think there's any actual problem here. It seems like this might just be a matter of the video itself of being of rather low quality. I had a quick look at the video on my computer as well, and it doesn't look much better. I can barely make out half the text on the screen. Consider this...
1) The video getting uploaded by a streamer is not going to be of particularly great quality to begin with, since it's generally getting encoded in realtime on their computer, while they are playing the game.
2) The resulting video file is getting re-encoded by whoever did the compilation video, reducing its quality further.
3) Youtube re-encodes their uploaded compilation video for efficient playback from their servers, reducing the quality even further.
4) The video is upscaled from 720p to 1080p to fit your screen, introducing some additional blur.
5) And then of course, you snapped a jpeg of your screen, resulting in another reduction in quality.
And that's assuming the video wasn't clipped out of some other prior compilation video, and re-encoded additional times. It seems to simply be a low-quality video at 720p, that's then scaled to fit your screen. You shouldn't expect it to look like a blu-ray, or even a television broadcast. The quality won't even be as good as if you were watching the gameplay when it was first streamed, since it's been re-encoded multiple times since then, introducing additional compression artifacts every step of the way.
Try watching a high image-quality youtube video that's available at higher resolutions, like this one for example, and see if there's still any problem...
https://www.youtube.com/watch?v=LXb3EKWsInQ
Even there, with high-quality source material, you should notice that there's a visible improvement when switching between 720p and 1080p on your 1080p screen. You should even notice a significant improvement setting the video to 1440p resolution, since Youtube adds a fair amount of compression to videos, making even 1080p content a little blurry on a 1080p screen.