It seems to me that mobile apps are using animated transitions in like websites used animated transitions in the early days of the internet, for a couple key reasons.

First, technologists are in awe of their ability to create new and interesting interactions. This self-adoration typically leads to creations that are simultaneously both amazing and annoying.

Second, with the relatively new technology, neither developers nor general audience users have defined a "normal" experience. Thus, animations will help provide context until industry navigation standards emerge.

Websites have standardized many activities over time based on experience. Thus, the need for animations and animated transitions in particular has declined to almost be non-existent. They are frequently used in tutorials. The desire to use them elsewhere has declined due to the risk of losing users. On successful websites, you don't want to "invent" an animation that loses users.

So many app animations are essentially a technology transitional necessary evil, in a sense. Until app designers develop industry standards for the new technology (apps and smartphones in general), animations are used in these limited cases:

1. where the app designer is not using standard navigation out of ignorance and is not aware of standard navigational options and techniques, the result of inexperience or limited resources

2. because the designer is challenged with something in their particular app that creates the need for a new type of transition or anchoring mechanism for good UX and animating it makes sense in order to keep users oriented. Because the technology was new a few years ago, there are many cases where this was necessary as described in other answers. This happens with new features like text messaging, which essentially did not exist before apps and vice versa. These will likely recede as the general audience becomes familiar with the technology and activity based expectations emerge.

3. the designer is so absorbed with themselves and their ability to do or create something, that they create and place animations in an app without regard for an existing, successful pattern that should be used. Or worse, they are trying to invent a better mousetrap (animation/transition) because they feel that existing successful patterns still are not good enough. This is just narcissism, but on rare occasion the narcissist is right and takes it to the next level.

It appears mobile apps use animated transitions like websites used animated transitions in the early days of the internet for a couple key reasons.

First, technologists are in awe of their ability to create new and interesting interactions. This self-adoration typically leads to creations that are simultaneously both amazing and annoying.

Second, with the relatively new technology, neither developers nor general audience users have defined a "normal" experience. Thus, animations will help provide context until industry navigation standards emerge.

Websites have standardized many activities over time based on experience. Thus, the need for animations and animated transitions in particular has declined to almost be non-existent. They are frequently used in tutorials. The desire to use them elsewhere has declined due to the risk of losing users. On successful websites, you don't want to "invent" an animation that loses users.

So many app animations are essentially a technology transitional necessary evil, in a sense. Until app designers develop industry standards for the new technology (apps and smartphones in general), animations are used in these limited cases:

1. where the app designer is not using standard navigation out of ignorance and is not aware of standard navigational options and techniques, the result of inexperience or limited resources

2. because the designer is challenged with something in their particular app that creates the need for a new type of transition or anchoring mechanism for good UX and animating it makes sense in order to keep users oriented. Because the technology was new a few years ago, there are many cases where this was necessary as described in other answers. This happens with new features like text messaging, which essentially did not exist before apps and vice versa. These will likely recede as the general audience becomes familiar with the technology and activity based expectations emerge.

3. the designer is so absorbed with themselves and their ability to do or create something, that they create and place animations in an app without regard for an existing, successful pattern that should be used. Or worse, they are trying to invent a better mousetrap (animation/transition) because they feel that existing successful patterns still are not good enough. This is just narcissism, but on rare occasion the narcissist is right and takes it to the next level.