[verb (dictionary form)] + ようになる [verb (potential)] + ようになる
ようになる says a change has come about: you can now do something you couldn't, or you now do something you didn't. It follows the dictionary form, and with a potential verb it means "become able to".
The change is to a state that carries on – a new ability or a new habit – rather than the moment something starts.
ようになる turns a verb into a change. An adjective or a noun changes with なる on its own, and so does a verb that already means a change.
ようになる is a change that happened, whether through effort or not. ようにする is the effort itself: making a point of doing something.
For something you no longer do or can no longer do, the ない-form changes to なくなる. ないようになる also exists, but sounds stiff.