A typical example are resources like some kind of database handler for example which needs some initialization steps and/or logic to clean up resources when it's no longer needed. In this case lifecycle callback methods called by the framework (IoC container) are a convenient way to initialize and destroy resources exactly once at the right time without manually having to call a init() or cleanup() method.
The use case would also vary by framework.
As an example in the Spring Batch Space - framework methods like @AfterStep, @BeforeStep are heavily used.
Internal bean life cycle are also used internally by the Spring Framework - e.g the PropertyPlaceholderConfigurer would need to implement BeanFactoryPostProcessor to inject properties before other beans are created.