How do you put a quote into context?

It means when you use a quote, you show what the original scenario was when the person said the Words. As an extreme example, if someone answered 'yes' to a question, without 'putting it in context', one would have no idea whether they are agreeing that sheep have eight legs or if London is the capital of the UK. So putting it in context means telling the background, in this case saying what the question was.

Taking quotes "out of context" is of course a popular technique in poor sensational reporting, where essentially a reporter could make another person seem to have said just about anything just by asking particular questions and then not revealing their context.