If you read the descriptions of those products, you will see phrases like "Rooting will be required depending on model". That is because you are not supposed to be able to take screenshots programmatically, though rooted device users may be able to pull this off, at least for some devices.
Another technique, that at least one such app uses, is to involve a developer computer in the process, using the mechanisms in Android used by DDMS and such for taking screenshots. In reality, it is the computer, not the device, taking the screenshots.
For the devices that such apps work on without requiring root and without requiring a developer computer, the authors of those apps are exploiting security holes in individual devices, just like a malware author would.