Jesus is making a point that He addressed on several occasions: He cares less about earthly appearances and more about who people are on the inside - He cares what we become.
In verse 35 He acknowledges that He's looking on the heart:
A good man out of the good treasure of the heart bringeth forth good things: and an evil man out of the evil treasure bringeth forth evil things.
And in verse 37 He further confirms that our words do matter:
For by thy words thou shalt be justified, and by thy words thou shalt be condemned.
How to reconcile these verses? This string of verses would be very disconnected if the message were about people judging the righteousness of other people. What person A says may indicate to person B what is in person A's heart, but Jesus' point is more than that.
God who sees what is in the heart (1 Samuel 16:7) and hears what is said in private doesn't need our public appearances in order to know how to judge us: we will be judged for who we really are on the inside--including who we are in private--regardless of how showy our public behavior may be.
But verse 37 makes it clear this isn't a license to sin - there will be no "I was a good person inside even though I disregarded God's commands". This may fool other imperfect humans but will not fool God. So words said in private are very much in scope here.
Is this hyperbole? I conclude it is not.
Target audience - everybody God wants to help become more than they are today (ie everybody)
Does this apply only to verbal utterance? No, if what really matters is what we become, any and all of what we are/think/say is applicable. Saying something verbally rather than putting it in writing may be a popular tactic in avoiding legal consequences in this life, but it will not prevent full and complete evidentiary discovery on judgement day.
How should believers take this? This may be more a matter for SE-Christianity, but it's worth asking ourselves "who am I becoming?" And "Not only in thoughts, words, and deeds, but in my innermost desires - am I more Christlike today than I was yesterday?"