Misconstrue means something is interpreted the wrong way. If you see your brother walking your best friend home, you may misconstrue the situation and think they're dating when really he's only picking up your birthday present.
The verb misconstrue originates from the words mis-, meaning "wrong," and construe, meaning "construction." Combined they mean "to put a wrong construction on" — in other words, something is given the wrong interpretation. You may misconstrue your brother's intentions when he takes your bike without asking. Sometimes people can intentionally allow others to misconstrue something for personal gain. A politician may let voters misconstrue his voting record to get elected.