Well, for one, it would help give them a cue that your civilization is not some pompous upstart who needs to be whittled down (through war), that you have real concerns they can respectfully accept or decline instead of black box demands that only serve to antagonize them.
And of course, it can also help tell you if this friendly civ is really interested in friendly relations and mutual co-existence and all that jazz.
Example: you start off next to Gandhi and Oda Nobunaga. Both appear friendly to you, so you try and request something from each of them that you need at the moment (gold, luxes, iron etc.) Gandhi is willing to help you by sparing 2 irons for free, while Oda would only give you a lump sum of 13 gold and nothing else. More importantly, Oda has settled a chain of cities in your direction. I imagine there would also be a dialogue/barter mechanic here that would add more flavor than just straight up requests, including random chitchat that might just give you positive modifiers with Oda (he likes your personality!) so he might not necessarily DOW you at all, or in Gandhi's case, he might ask you why you really need that iron for urgently and thus may not give in, but that's a little off the scope for this.
