Assume can mean
to take over the responsibilities,to take upon one's self.
-To assume the office of treasurer
It means to rush to judgement or come to a conclusion about a situation or someone without knowing any facts before hand about the matter. Usually the person who *assumes* is wrong/incorrect.
Example:
He assumed I ate the last apple because I was the only one in the house, but it was actually our younger brother.
He assumes we are talking about him because we are whispering, but we are talking about a joke.
They assumed I was mute because I did not speak during class, but actually I am shy person.