From the manual:

Influence - Influence is a crucial resource for dealing with other nations and creatures of the world. It can be used directly as a currency when trading with other nations, as well as altering the value of your goods in trades with others. In addition, when building camps to recruit wild creatures, such a dragons and ogres, you will need influence to get your construction underway. Influence can be gained through city improvements and resource nodes in the world.

This is copied from something I posted on another website when this was asked.

It has a few uses and is more useful depending on what leader/faction you are playing. For the average nation, you can use it to build things like ogre camps, mercenary camps, dragon lairs, those things if you have researched them and have the map resource. Also, you can use them to trade for more useful stuff from the ai, like gold, metal, etc. Also, if you want to build Sions, you will need 100 influence to do so. Sions are truly difficult to get any use out of, and I wouldn't bother until you really know the game well, and even then it's iffy.

Now, some specific things you can do with influence based on your leader or faction.

If you are playing Kraxis, or custom made faction with Betrayer, you can use 500 influence to steal another factions city from them. I have never done this, I have no idea how well it works.

If you are playing as Altar, or a custom faction with the race of "men", than you need 50 influence to build each Henchman, a very useful unit, kind o fa mini-champion.

If you are playing a leader that has the Diplomat profession, like Karavox, than you can spend 30 influence in battle to try to make a enemy trained unit defect to your side using the silver tongue ability. This can be a fun mechanic.

It's boring.

Needs a complete overhaul imo.

• Heroes should cost influence to hire, not gildar. Additional heroes should cost incrementally increased influence.
• Empire growth should be a dependant factor based on how much influence you have.

I put together a suggestion proposal that can be found here: https://forums.elementalgame.com/433096

Thanks. I forgot about wildings and such costing influence. I didn't know that henchmen cost influence, so that is good to know. I was thinking that influence might have some correlation to diplomatic relations, but had no idea how that worked. I am going to assume it does not, and only worry about keeping enough to hire henchmen.

Like the ideas, Firefly. Influence is kind of a "tack on" mechanic. I'd love to see it more integrated earlier on. You have to be very intentional about finding/using it, and I think the role could be expanded and integrated like you mention.

It almost makes me wonder why there's both Prestige and Influence. They kind of mean the same thing and could be combined to work together. Prestige is also a hard one to get to use/understand.

This is a great idea, and makes logical sense.

The funny thing is, that Influence is suppose to be traded to the AI for peace treaties, materials and shit like that.  But, they don' value it very high.  I think Influence is worth 2 gold to them.  But if you need Influence to build a resource or henchman and want to trade for the influence they have, it cost 100 gold per influence.  It's pretty fucking ridiculous.  Also, unless you have the Heroic faction trait, you will never have very much Influence.  So this fucks Kraxis the most, as they have two special abilities that need influence and no way to build it.  Altar on the other hand, needs 50 per henchman, but has heroic and wanderlust so they can get thousands of influence and can build dozens of Henchmen.  

It doesn't make sense to me for influence to be resource that can be gathered and spent like currency.  It would make more sense as a sliding scale that affects your diplomatic relations and overall "health" of your empire.

I think it would be great if influence would affect a city's defenses. 

- Rallying Cry. Sway the locals to your cause. This adds a minor bonus to a city defender's attack/defense/hp.

- Last Stand. This adds a moderate bonus to a city defender's attack/defense/hp, and is castable/usable once per city.

The more that I think about it, the more I think that influence could be used to implement espionage.

- Desertion. An opponent within your borders has random units leave his party or return to his nearest town.

- Safety in Numbers. Cast on a unit (sovereign/champion) or party. This inflates the party's combat rating for a duration. 

- Natural Diplomacy. Wandering creature wanders away from your borders.

- Natural Guidance. Wandering creature moves toward an opponent. The player pick's the opponent or opposing city, and the creature's liklihood to reach its destination is affected by the spell caster's level, creature's type and level.

Edit: 'Rumored Threat' renamed 'Safety in Numbers'.

Champions should have a salary---but a powerful champion should actually start adding influence.

I'm not sure about it but the AI seems to consider the amount of influence you have/get per turn and this accounts as a diplomatic bonus in foreign relations with other kingdom (+2).

There is a mod which does exactly this. You can spend influence to spy and counterspy, etc. I've played with it a bit, and it seems to work OK. It's called Espionage and is at v0.91 as I write this (https://www.wincustomize.com/explore/elemental_fallen_enchantress#/page/2)

How do we propose this - an excellent suggestion

This needs to happen. It's a perfect way to balance out the heroes.