IPB actually had this back at 1.3 or something(early) but decided to remove it for some reason. I have been in situations countless times where I would've needed something like that.
Essentially, part quote does exactly what it says it does, it quotes a part of a post. More specifically, it quotes the part you select. So I can select a sentence of a post or a whole segment, click the button "part-quote" and it will quote that part that I selected - with correct linking and the name aswell.
So let's say you select the first segment of this post, clicking part-quote will quote that part aswell as link to my post and saying "Tanax said", just like a regular quoted message. This will allow easier quoting when you want to reply to different parts of a post(and also over several posts). Instead of clicking "reply" and then manually have to "chop" that quote up to pieces before being able to reply to the individual parts.