VS + 게 되다
Verb Stem + 게 되다 means "to start to -" or "to come to -".
The first sense, "to start to -", means that there has been some kind of a change and something is now behaving or doing something differently than it used to. This is usually used with verbs that imply an action rather than a state of being.
If you put it into the progressive form (VS + 고 있다), then it means something is starting to do something and will do more and more of it in the future.
The second sense, "to come to -", means that there was some sort of a change but it happened not because of any direct intention on the part of the actor. The universe was just in that particular state that things aligned and happened and now things "came to" be the way they are, without any particular blame on anyone's shoulders. The progressive form of this means that the process of this unintentional change is still ongoing. This sense is used with verbs that show a state of being or an attribute.
The roots of this phrase are VS + 게 (adverb clause) + 되다 (to become). You should think of it as the combination of these two forms, because it will help you truly understand what those two forms really mean.
See Also
G + 지다, which means roughly the same thing.