It is an unfortunate fact of life that the world is filled to the brim of what the Bible describes as "fools". Fools are people who don't particularly care about reality, and because of this, it's usually rather fruitless to argue with them. As mentioned in the header image, you can't use logic and reason against illogical and unreasonable people.
That doesn't mean you can't use logic and reason at all though. You don't have to play their games, but you can play a few tricks of your own. Jesus exemplified this when He would publicly embarrass the Pharisees, the scribes, the chief priests, the elders, and anyone else who tried to trip Him up.
One example of this is when Jesus was questioned about taxes in Matthew 22:15-22. The Pharisees, seeking to trip Jesus up, asked if it was lawful to pay taxes to Caesar. This was a catch 22 situation. If He said yes, He'd have made Himself very unpopular with the Jews. If He said no, He'd have been attacked by the Romans for inciting treason. But Jesus took option 3. Rather than playing their game, He asked for a denarius, showed that it had Caesar's inscription on it, and said that basically, the money belongs to Caesar anyway, so give it to him, but give to God what belongs to God.
The crowds could not answer Him. They tried to trip Him, but He instead tripped them. This is what it means to not answer a fool by his folly, but to instead answer him according to his folly. It basically means don't play their games, answer them as their foolishness requires.