to twist something out of true: to bend a straight path, a message, or a person away from where it ought to run
The picture underneath this word is physical. It joins the everyday Greek verb for turning (στρέφω) to a prefix meaning "all the way through," so it names a thorough twisting: taking something that ran straight and bending it out of line. Greek writers outside the Bible used it for a warped plank, a distorted face, and a crooked argument, and the New Testament keeps that image while applying it to people, to roads, and to teaching.
The one image is what lets a single word cover such different objects. It can name the act of bending, the thing that gets bent, or the condition left behind once the bending is done. A man can be turned off the road he was walking, a road can itself be made to zigzag, and a whole generation can be spoken of as already crooked. In every case the word assumes there is a straight line to depart from, which is why it is always a verdict and never a neutral word for change: this differs from simply turning away (ἀποστρέφω), which reports a change of direction and leaves the rightness of it open.
Its harshest uses borrow their language from Moses. Deuteronomy 32:5 calls Israel in the wilderness a crooked and twisted generation, and the Greek Old Testament used this very verb there. So when Jesus says "faithless and twisted generation" in Matthew 17:17 and Luke 9:41, and when Paul tells the Philippians that they shine in the middle of a crooked (σκολιός) and twisted generation, both are reaching back to that old song and setting their own hearers inside it. In these places the word has stopped describing an action and started describing a settled state: not people being bent, but people who are bent, the twisting long since finished. A close cousin, the verb for being turned inside out (ἐκστρέφω), is used the same way in Titus 3:11.