To put words into somebody's mouth is to claim that they said something they really did not. It usually happens when somebody misinterprets what a person said. "I don't like her as a boss." "You're saying she's too old." "Don't put words in my mouth! I never said that. It's not her age that's the problem - it's just her management style."

To put words in (to) someone's mouth is interpreting what you think someone wishes to express.