It means perform a process. Starting, middle, and completion. "Carry out" may be a better way to put it because it typically is used for the act of making a law or order happen, not for some natural act like giving birth. This is why many governments have an executive branch that carries out the laws made by the legislative branch.

It became used for the act of killing a prisoner because it was executing the legal order for them to receive the death penalty. It became used for running a program because the processor executed the "instructions" which is what they called lines of code.