"One-click" root apps don't work anymore. Why not?

Years ago, it was easy to root any device. You installed KingRoot, KingoRoot, or a similar app. You hit the large "Root" button, and let the app do the rest. After 5 minutes or so, a green "Root Succeeded!" prompt would pop up. You could choose to keep the root app; or you could uninstall it and install a clean root manager app like SuperSU or Superuser. * How do these apps work? * Why don't they work anymore? * If they do work, should I use them or not?