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Here is an example of how it works. An app developer has an app that needs to send you a push notification. Play Services has two APIs for that very function here. The developer uses those APIs in their app. From there, Play Services sends push notifications when the app asks it to without any additional work from the developer.
There are dozens, if not hundreds, of other APIs that help developers do simple tasks in a uniform way. Navigation apps asking for your location probably use Play Services to do it. Think of it as your phone’s manager, sending you notifications, giving apps your location, and running processes in the background for optimal battery efficiency.