Why does moral progress feel preachy and annoying?

It doesn't, to sane adults

Progressive ideas have always been deemed preachy and annoying. Conservative ideas have always been deemed preachy and annoying. Having to hear a different ideological opinion is going to perceived that way.

The internet has just made this attitude more visible. The internet keeps people in their own personal bubble of content and people that share views. But it also gives access for any opposing views be public, widespread, and benefits from positive or negative interaction equally. Where the most vocal and controversial make it to the front page of everyone’s feeds.