Always use this option with headings. It looks bad if a heading appears at the bottom of a page and the text it’s introducing starts on the next page. Keeping with the next paragraph prevents this.
The keep-with-next-paragraph option is a little boring on its own. It gets more interesting when used with its friend, the keep-lines-together option. For instance, I once had to prepare a jury-instruction form with many entries like this:
_____ Given as proposed
_____ Given as modified
Here, the name of the jury instruction is one paragraph, and the four choices below are a second paragraph. The four choices won’t get separated from each other because they’re glued together with the keep-lines-together option. But we don’t want the instruction name getting separated from the choices either. By setting the instruction name to keep with next paragraph, all five lines are guaranteed to move as a unit.
Why didn’t I make the whole block one paragraph? So I could apply a separate paragraph style to the names of the instructions, and reformat all of them as a group. (See paragraph and character styles.)