Why are military drones shaped so strangely?
The whale-shaped forward fuselage covers a parabolic antenna for a high-bandwidth datalink. The operators want to receive the reconnaissance data in real time, and by giving the antenna the best place in the aircraft, it will be able to connect to communications satellites even when they are just above the horizon.



Global Hawk cut-away drawing (picture source)

The V-tail of the Predator is an inheritance from its origins as a Navy drone which could be folded to fit into a torpedo tube. Read all about it in this answer. If you want more info on the benefits of an inverted V-tail, make sure you read KeithS's answer, too.

In the Global Hawk the V-tail was chosen to mount the engine above the fuselage for better IR shielding from below. The two tail surfaces also shield the exhaust from the side without affecting aerodynamics negatively.

The Aerosonde's configuration was chosen after Tad McGeer had experienced in 1990 from the 1/5th scale RC model of the Perseus UAV of Aurora that a tail-mounted propeller would stabilize the aircraft to the point that it became practically uncontrollable. A pusher propeller is more efficient than a tractor propeller, which also explains the propeller location on the Predator.

If it strikes you how many of the UAVs have V-tails: This allows to use fewer components (surfaces, connectors, actuators), and since all are moved by computer control, the issue of a mechanical mixer does not exist. The lower damping of V-tails can also be easily tolerated, since the FCS can react much more quickly and commensurately to disturbances than any human pilot. The lower maneuverability possible with V-tails is also not an issue - reconnaissance UAVs fly mostly straight and level.

The high aspect ratio wings are helping in extending the range and especially the flying time of observation drones. Similar to glider wings, the long and slender wing will offer the lowest drag for a given lift at subsonic speed.

In the end, each detail has been chosen deliberately and can be well explained. There is no conspiracy to make UAVs look different from Cessnas.