To misconstrue is to get the wrong idea about something. If you misconstrue a friend’s silence, you might get mad at him for ignoring you — until you find out he has a sore throat and can’t talk.
The verb misconstrue comes from mis- meaning "wrong" and construe meaning "construction." Combined they mean "to put a wrong construction on" — in other words, to interpret in the wrong way. Sometimes people can intentionally allow others to misconstrue something for personal gain. For example, a politician may let voters misconstrue her voting record if she believes doing so will help her get elected.