Early speculation (and it is speculation) has focused on a shift of the cargo inside the plane, which was reportedly carrying several military vehicles. How could some cargo shifting around possibly cause such an utter catastrophe?
Consider this simple model of how an airplane stays under longitudinal (pitch) control:
This is, of course, a vast oversimplification of the Queen of the Skies, but as a model it provides great intuition. Start with CL - the center of lift. This is the point along the airplane's longitudinal axis where lift, the upward force generated by the wing plowing through the air at an angle, is effectively concentrated. If you can imagine the airplane as a seesaw, this is the fulcrum. CL is largely determined by the physical shape of the airplane and the wing's angle of attack.
CG, the center of gravity, is the point where the weight of the airplane is effectively concentrated. In the above diagram, which represents a normal flight condition, CG is located forward of CL. In isolation, this results in a nose-down torque about the CL. To keep the airplane stable, this is counterbalanced by a nose-up torque imposed by the tailplane, the horizontal stabilizer on the tail, which is plowing through the air just like the wing, but at a different angle.
The nose-up torque of the little tailplane can counteract that of the airplane's immense weight, because it's so much farther away from the CL - leverage. That tailplane has a Goldilocks design: large enough to counteract the torque generated by the weight, but also minimizing the additional drag it generates as it plows through the air, and the additional lift needed from the wing to maintain altitude. The pilots' primary flight controls can adjust the torque applied by the tailplane, but only within certain design limits.
Importantly, CG is heavily influenced by the distribution of stuff inside the plane - fuel, people, and cargo. If the CG is too far forward, the tailplane won't be able to supply enough opposing torque to keep the nose up. If the CG is a bit too close to the CL, the tailplane's torque becomes disproportionately large compared to the weight's - the controls will become unusually sensitive, possibly leading to pilot-induced oscillations. If the CG is actually behind (aft of) the CL, the tailplane would have to generate its own sustained lift in order to keep the plane under control - something it was never designed to do. Past a certain point, the airplane would go into an uncontrollable nose-up attitude. This is consistent with what we see in the first few seconds of the video.
During preflight planning, the pilot-in-command is responsible for ensuring that the CG is within the allowable range and will remain so throughout the flight. I do these calculations myself before I fly a little sport plane, assisted by diagrams like these:
What happens then? The nose rises and the primary flight controls are ineffective in lowering it. The uncontrollability actually worsens with the wing's increased angle of attack. Momentum keeps the plane climbing for a while, but it quickly loses speed in this frighteningly steep climb - the engines aren't nearly powerful enough to propel the airplane almost vertically against both its own weight and the drag of the enormous wing and belly now plowing largely forward. What lift the wing still generates then disappears when it stalls. The plane falls, and recovery is impossible with only a thousand or so feet of altitude.
A mere broken tiedown in the cargo hold could have set off this tragic chain reaction. And but for the timing, the results might have been far less disastrous.
All speculation, but simultaneously fascinating and terrible to contemplate. There are numerous other possible causes and contributing factors, including severe weather in the area at the time, and the steep takeoffs typical at Bagram to avoid potential threats. The investigation will tell.