From the game's encyclopedia
Influence represents a Character's political reach and authority in a Realm that has the Administrative Government.
Influence is an essential tool and can be acquired in numerous ways. For example by becoming a Governor, upgrading an Estate, or being allied to the House Head of a Noble Family.
The main way to spend and make use of Influence is to acquire a Governorship, but is also a valuable resource to sway other Characters to do your bidding.
Every time a Character gains Influence, they also progress towards their next Level of Influence.
The game's text for patch 1.20.1.2. Words in blue are entries of the encyclopedia: hover one to read it, as in the game.
Influence is an entry of Crusader Kings III's in-game encyclopedia. The game explains it: "Influence represents a Character's political reach and authority in a Realm that has the Administrative Government." It is part of Administrative. It covers Level of Influence. The game also links it as Influential. Patch 1.20.1.2, from game/common/game_concepts/07_ep3_game_concepts.txt and game/localization/english/game_concepts_l_english.yml.
Source: game/common/game_concepts/07_ep3_game_concepts.txt, game/localization/english/game_concepts_l_english.yml and the English localisation, Crusader Kings III 1.20.1.2.