The Search Box is basically an all-purpose search box.
It can search the current document, all open documents, menu items, help items, and, if enabled (as sending queries online is a privacy concern), online help items, tutorials, and commands from downloadable extensions. If the list of search results is too big, it’ll get a scroll bar.
When you push one of the “find” arrows (the ones you use to search the document), all of the command suggestions will get hidden under an expandable menu so that you would see more of the document.
(Oh, and it’s accessible with a keyboard shortcut. And you can use the arrow keys along with enter to navigate through it.)
Click or Tap 
Interesting approach. I wonder however if it shouldn’t be in a second status bar (i.e. at the bottom) like it is in Firefox. It would have a similar feeling and would also work with a narrow window.
More seriously, one thing sorely missing in LibreOffice and in competing products is the way to search easily for paragraph, new line and other formatting marks. I would suggest a window “enlargement” where one would click the formatting features one want to search for.
“I wonder however if it shouldn’t be in a second status bar (i.e. at the bottom) like it is in Firefox. It would have a similar feeling and would also work with a narrow window.”
The idea of the Search Box is to basically have one central box for searching through everything and anything. There could be a bar for searching the document, but the same functionality would stay in the Search Box.
“I would suggest a window “enlargement” where one would click the formatting features one want to search for.””
All this would be available under the “More options” dropdown.
I’m against having it in a status bar. It will mainly be used to search functionality hidden in one of the many menus, so it belongs *somewhere near* the menus imho.
Wow, this is very nice and useful!
See also this feature request: https://bugs.documentfoundation.org/show_bug.cgi?id=91874