How do I display the messages under the discussion topics?

Hi!

I can't figure out which plugin or setting enables previewing of posts in the topic like on the main Flarum?

Like here

Hello, it is this extension that allows that.

Without any extension, Flarum does the following:

Show excerpt of the first post for pinned discussions
Show text of first matched occurrence of the text in search (that's what we see in your screenshot)

As ChadJessen points out, there's an extension to enable the behavior of pinned posts for all posts.