They want you to use their mobile app if you want to view YouTube. You could assume this is because they made an app specialized for YouTube, but then people still just use their mobile browser. Since a mobile browser can never be 100% fine with a Desktop version of a site, whether it be bug or simply not fitting, they also have to manage a mobile version of the website. Forcing you to use their app makes it easier on them, while also allowing them to force advertisements and such.
It's a lot of different things, both logical and assholic. Of course as they mentioned, Desktop Mode should bypass this as trickery. And making your videos not viewable on mobile browsers is actually a setting which video uploaders can choose not to obey, but then they get certain penalties in terms of their own monetizing and such... So many things.