You are standing by train tracks and you see a train going down about to hit a stalled bus filled with children. You are right near a switch that can divert the train, but there is a man standing on the other track. Most people answer that they would divert the train. The reasoning is that the man has a better chance of running off the track than all those kids on the bus, and why not save the most lives possible. Now what if you were on a bridge above the tracks and saw that train on its way to hit that bus but the only way you could stop it was to push an overweight man standing on the bridge over to stop the train. Surely the man would die, but you would save all those kids. Most people answer that they could never do that. The certainty of that man's death by ones own hand is too much to bear, even to save the lives of small children.
Now new drone technology has made it possible to follow a cell phone signal and send a missile directly to a terrorist's car without so much as damaging the vehicle next to it on the road, and Rand Paul and many GOP Senators are outraged?
Perhaps their reasoning is similar to the train allegory. A bomb that is dropped may kill hundreds or thousands, but it can also miss its target entirely. That possibility makes it psychologically more palpatable to deal with. A drone on the other hand is personal, immediate, and very effective. It is controlled by a person, and every step is recorded. Shots are fired from an unfair vantage point high in the sky. There is little chance of escape once a drone focuses in on its target.
This thought of a focused killing without capture, without trial, is what has the GOP and Rand Paul all upset. The army could drop thousand of bombs, but one tactical strike ordered by the President and Rand Paul will filibuster for hours on end. He is standing up as a defender of the Bill of Rights.
The other side sees the Bill of Rights as a document written to save lives. If the document needs a little tweaking to save lives, so be it. If we need to actually kill someone with our own hands, to save the lives of many, so be it.