Following the release of iOS 7 (and iOS 8 as well), animation and motion effects became central to the design of apps from both Apple and 3rd party developers. iOS 7 introduced a flat, minimal design to apps which inevitably resulted in some apps having a similar UI. To distinguish their apps from other apps, developers employed such features as animations and motion effects to make their apps stand out.
Not only are animations used to set your app apart, they can improve the overall user experience of your application. For great examples of how animations are used to improve UX, you should look at how Apple uses animations in their apps. For example, in the Photos app, when you select a photo from a collection, the photo expands out from the selected one and on closing it, it shrinks back to the selected photo. This adds to the navigation of the app in that it lets you know exactly where you were, if browsing many pictures.
Facebook’s Paper also employs animations beautifully to add to the overall user experience of the app. You select an article to read by flipping it up. This action, the fact that the article expands out from its thumbnail version, suggests that in doing the opposite, i.e. flipping the article downwards, would shrink it back to its thumbnail. Here, animation is used to convey how the app works and even a first time user of the app would soon be able to make assumptions on its use and figure out its navigation without needing a tutorial.
Not only do animations add to the user experience of the app, but they could be what delights and impresses your users guaranteeing repeat usage instead of uninstalling the app for a better one from the App Store.