The following materials were used, while compiling this guide:
http://www.twcenter.net/forums/showthread.php?776322-Diplomatic-mission-outcome-determination
https://steamproxy.net/app/214950/discussions/0/1698293255134300673/

I am not a dev/modder so I supplemented this knowledge with some testing and compiled it into one guide. Few open questions still remain though.

In general this functionality should be utilized to rise your relation with other factions. This might be a bit problematic, as it's not properly described anywhere. Only the immediate impact on Political Parties is listed on the tooltip which is nice, but insufficient.

There are few main factors, that determine the 'send diplomat' result:
1. The diplomatic relations between your and the target faction (the better relations the bigger the positive impact on the outcome).
2. Cities ratio:
“The number of cities of the diplomat’s faction” divided by “The number of cities of the target faction”. The more you have the better.
3. Gravitas of the diplomat. As before, the more the better.
4.(*see below) Political ambition of the diplomat - the higher the better.

After all those factors have been taken into account(let's say they have been quantified and added to each other), then a random number is generated and compared to the added factors.

There are four possible outcomes:
-Critical Failure: ending with diplomat death;
-Failure: diplomat might get wounded or cause some faction-wide debuffs. Diplomat gains 5 gravitas;
-Success: you will get either 5000 money or a faction buff. Diplomat gains 10 gravitas.
-Critical success (possible only if the target faction has more than one city and is bordering you(**see below)): You will gain one settlement of the target faction. Diplomat gets 20 gravitas.

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* - I seen this mentioned in reply to one post, but was not able to confirm it yet.
** - I have not tested this extensively. Supposedly, the border requirement was patched in at some point. There is also an open question if only border settlements can be given or any settlement of the bordering faction.

The 'send diplomat' option can be useful as a 'supplement' to your diplomatic actions, rather than a replacement for 'throwing gold' or 'fighting the enemies' of the faction you are trying to influence.

If a faction had conqured a piece of land that you need, you might want to regain it via political action, altough the settlment that you woudl get might be different than the one you want.

If you are playing with client-states/allies you might utilize this mechanic to regain territories that were conquered by your allies, that you actually need.

Finally, your main way of influencing the outcome is to send characters with high gravitas. I usually gain those by demobilizing old generals or having multiple character promoted few levels.

Is this a game defining mechainc? No.
Is it easy and fun to use? Not really.

Then why bother? It allows for role-play, some additional boost to faction loyalty, ability to gain some buffs or money (via a gamble, but still) and to gain setlements without waging war.

Reccomended MOD (it stops your diplomats from dying):
https://steamproxy.net/sharedfiles/filedetails/?id=1397295936

Small note, possibly for modders: There is very little missing for this to become a fun and importnat action. If you could choose to negotiate a buying/gaining of settlement, and mark a particular settlement, this would be a far more importnat and engaging mechanic. Combine this with better explanation and a periodical reduction in success chance after a critical success and you would gain a great way to both role-play and increase the value of your characters.

Since I am not a programmer, I could use some help with the list below, as testing this might take some time and not yeald 100% accurate results.

1. Confirm that only settlements of bordering factions can be given to player.
2. Confirm if only a settlement that is bordering the player may be given to the player.
3. Confirm if the AI is using this feature (or is able to).
4. Check if the Political Ambition is a factor and what is it's impact.
5. Gather the list of buffs and debuffs.