벗어나다 means to (intentionally) free oneself from something or to (accidentally) stray/deviate from an intended course. Your example is using it in the second meaning. "to be distinct" is a misleading definition.
남녀의 차이를 인정해야 한다는 작가의 메시지에서 벗어나 점점 남성들을 비난하는 분위기가 되었다.
= Increasingly straying from the author's message that we should acknowledge the gender differences, it turned into a bout/atmosphere of men bashing.
* [Subject] ...XX-(아/어)서 ...YY-가 되었다 = [Subject] has XX-ed, leading to/resulting in YY. (서 can be dropped, as in your example: 벗어나다 + -아/어서 -> 벗어나서 -> 벗어나).
The sentence has no subject. It must be referring to what was just mentioned, perhaps some meeting or discussion.