Diplomatic Visibility allows you to receive Gossip, information about another player's actions. You gain more detailed Gossip based on your level of visibility, of which there are 5 levels: None, Limited, Open, Secret, and Top Secret. For example, when you have just met a civilization, you will receive information about their declarations of war, denouncements, friendships, alliances, and when they do something like found a Pantheon or complete a wonder. If you gain Limited Diplomatic Visibility, you will find out some information about their opinion of you and what government they have. At the Open level, you will have access to that civilization's diplomatic agendas, their relationships with other civilizations, and know when they have launched an inquisition.
You must accumulate Diplomatic Visibility levels individually for each civilization. Levels may be gained by accumulating various sources of Gossip, including:
Traders. Establish a

Trade Route to a civilization to increase visibility by one level.
Send a Delegation. Send a Delegation to a civilization to increase visibility by one level. Once Embassies are available, establishing an Embassy will replace this.
Send a spy on a Listening Post Mission. Performing the Listening Post mission in another civilization's city increases visibility by one level.
Forming an Alliance. Alternatively to spying, you can form an alliance with a friendly civilization. These two actions do not add separate Diplomatic Visibility levels - it does no good to spy on your allies!
Research the Printing Press technology. This will increase your visibility with ALL civilizations by one level.
Play as Catherine de Medici. She gives France one level of access above normal at all times with civilizations she has met.
When you have Secret or Top Secret level visibility on a foreign civilization you can click on one of that civilization's city banners and view the City Details screen for that foreign city.