Key-value pairs (a.k.a. name value pairs) allow the app publisher to pass along special instructions for the app along with a push notification. This functionality is commonly used for deep linking users into particular parts of the app after they have opened a push notification.
A news app, for example, may send a breaking news alert via a push notification, and it might include a key named "Article ID" and a value of "10042" along with that push. This relies on the app developer having encoded logic into the app that understands the "Article ID" key and how to handle the value appropriately.