Well, one example I like when people ask me "what's contravariant good for?" is the following intuition. Suppose we're doing stream processing. We might have a datatype 'Source a' which is a source that produces a stream of a's.
'Source' is an example of a Functor. We can use 'fmap :: Functor f => (a -> b) -> f a -> f b' as '(a -> b) -> Source a -> Source b'. So, if we a function 'a -> b' we can turn a source of a's into a source of b's. Nice.
In such a library we would obviously also want a 'Sink a', this is a datatype that consumes a stream of a's. For example, it might write these a's to disk. Now, clearly it doesn't make sense for 'Sink' to be a functor. Think about it, if we have a sink that writes a's to disk, how would a function 'a -> b' affect it? Sure, we could turn all a's into b's, but then what? We don't know how to do anything with b's.
However, 'Sink' is a Contravariant (Functor). So, let's have a look at that. 'contramap :: Contravariant f => (a -> b) -> f b -> f a', so '(a -> b) -> Sink b -> Sink a'. If we have a 'Sink' that writes 'b' to disk and a function that turns a's into b's we can obviously construct a 'Sink a' that consumes a stream of a's, converts them to b's and passes them to the original 'Sink'.
And then there's a third abstraction not mentioned in the original post. The 'Profunctor', a Profunctor is a type that has two arguments and is contravariant in the first one, while the second is a regular Functor. In other words, if we have 'Pipe a b' this type can be made a Profunctor which comes with 'dimap :: Profunctor p => (a -> b) -> (c -> d) -> p b c -> p a d', which hopefully looks very similar to both Functor and Contravariant. In our stream processing example 'Pipe a b' would correspond with a pipe that consumes a stream of a's and turns it into a stream of b's which we can use to plumb 'Sink' and 'Source' together. We can both contramap it's first type argument to change what we can feed into it, as well as fmap the second type argument to alter what it produces.
These are from the only cases where these classes show up, but I hope they give some relatively easy to follow example on how these classes can capture some common scenario.
