So, imagine you have a really bad plant in your garden. It's growing out of control and taking over all the other plants. You want to get rid of it, but just cutting off the top won't do the trick. You need to dig it up by the roots and make sure it can never come back. That's basically what "terminate with extreme prejudice" means.
But instead of a plant, we're talking about people. If someone is causing a lot of trouble and doing really bad things, sometimes the only way to stop them is to "terminate" them. But not just by killing them quickly, we mean make sure they can never do anything bad again. This is where the "extreme prejudice" part comes in. It means using a lot of force and making sure the person is completely stopped, like pulling up all the roots of a bad plant. But it's a big decision to make and should only be done in very rare and extreme circumstances.