Since I wasn't satisfied with the answers here, I did some studying, and I thing I can explain, what enables the torque multiplication.
I will follow the fluid starting from entering the pump, and explain how it will apply torque to each of the parts.
First, the fluid will enter pump at the center, which will both move the fluid through through the pump, to the outer edge, as well as add rotation to the fluid in the direction of the pump. This will naturally apply a torque to the pump in the direction opposite to the rotation of the pump (this torque is the torque coming from the engine).
Now, the rotating fluid will move from the outer edge of the pump to the outer edge of the turbine and enter the turbine. In the turbine it's blades are shaped to reverse the rotation of the incoming fluid, while it is pushed to the center. This interaction will apply a torque to the turbine (and from there to the transmission) in the direction of the rotation of the incoming fluid (opposite to engine torque).
Last, comes the critical part to the torque multiplication, the stator. The stator is in the center of the torque converter, between the turbine and the pump. The stator blades will once again reverse the rotation of the fluid coming from the turbine, to match the rotation of the pump. This will apply a torque to the stator in the direction opposite of the one to the turbine (the stator is fixed in place, though, as it's name tells us, so it will not rotate). Exiting the stator, the fluid will repeat it's cycle.
So, what is it, that allows the mismatch between the input torque and output torque. It is the torque applied to the stator. At an equilibrium, all the torques in the system will add to zero, so the torque applied to the engine must be exactly the opposite of the sum of the torques applied to the stator and the output. Since the output torque and the stator torque are opposite, the output torque has to be larger than the input torque (by the torque applied to the stator to be exact)
Hope this helps someone to understand the black magic of torque multiplication. For me, at least, it was a mystery for far longer than I'm willing to admit.