its not about rigging elections - its about coup - ing
"Election rigging - Stationing a spy in a City-State will set him to rig the local elections which occur regularly every 12 turns. If he's successful, the new City-State government will be more friendly to your civilization, and less so to others, meaning you'll gain Influence with the City-State, while others will lose it. Enemy spies may also be acting in the same City-State, and then the spy with better experience will have better chance to rig elections. Other bonuses may improve chances at rigging elections.
Coup d'état - While in a City-State, your spy may also stage a coup. This is only possible when the City-State already has an ally. It is a dangerous course of action, because you risk losing your spy if the coup is unsuccessful. However, if successful, the City-State immediately allies itself with you, while their previous ally's Influence is reduced to the level of Influence you previously had on the City-State. The success rate of the coup tends to depend on your spy's experience level, the presence of other spies in the City-State, and your current Influence level with them (the higher your Influence, the more chance of success, because your spy may use the existing sympatizers to bolster the coup attempt).
Staging a coup may be the only way of making a City-State an ally when their current ally has enormous Influence and you have no practical hope of eclipsing it. It is also the only way of getting out of war with the City-State if you're already at war with their current ally. "
