But pure also became the word that was used for grain that had been processed in order to remove the chaff from the pure wheat. I used this illustration a few weeks ago. It's like when you whip egg whites. Unless the egg whites are pure they won't whip. Even the smallest amount of yolk will stop them from whipping up.
Purity describes freedom from anything that's false or insincere.
It's the opposite of hypocrisy, being 2-faced. It's similar to being single-minded which Jesus talks about later in the Sermon on the Mount when he warns against trying to serve both God and wealth. You see the trouble with us is that so often our hearts are divided. One half wants to obey God while the other half resists him. It's like there's a fold down the middle.