The influence is a value that affects the ways the player may interact with a given location on the Map.
To gain influence in a city you have to make sure that guests hailing from there will leave the inn satisfied. In the map window, if you left-click on a location, you can choose a magnifying glass icon - in the window that appears you can learn more details about the place as well as use political (blue) gossip currency to increase your influence.
The bar below shows what your current level of influence does (it may allow you to create a trade route, increase the quality of goods, allow learning recipes from the location, etc.)
If you have enough influence to create a trade route, choose this option from the available actions that appear upon clicking the location with the left mouse button. To create a trade route you can spend either money or the economic (green) gossip currency. Creating a trade route unlocks new locations, near those now-available for trading.