So basically it's a Tiger 1 chassis that is converted over to be a TD like looking vehicle with a large rocket launcher cannon on it. The thing weighed 68tons which is half a ton lighter than the Tiger 2 and 14tons lighter than the Tiger 1. The thing with the Tiger 1 is that the engine is underpowered and it suffered mechanical break downs. So this thing would be even worse due to the large weight on the problem chassis which is being powered by an underpowered engine. It was designed from 1943-1944 and it was being produced from 1943(4)-1945. Only 19 of these things were built.
For the  380 mm rockets they are a two stage device. They get sent out of the cannon and then the rocket engine takes over. If you look at the front of the cannon you see all these holes in a circle going around it. Those are for the gases. The rounds are so heavy that the only way to load the thing is from the crane on the side. Once inside the entire 5 man crew had to load the rounds into the breach. The vehicle was able to carry 14 of those 828lbs monsters. One already loaded and 13 others in the ammo rakes.
For combat as the name "Assault Tiger" suggest they were made to be attacking enemy fortifications with the support of the infantry. They could be back at about 20,000ft to fire their explosive charge of 125 kg(276 lb) at their target. The thing though about this due to the year they were produced Germany was on the defensive. So these things would have been used more for a defensive role than a attacking one.
For combat the limited numbers of the thing saw action during the Warsaw Uprising, the Battle of the Bulge and the Battle of the Reichswald. There's a story about an AT destroying Sherman tanks in one rocket. Could it happen? Maybe if the tanks are close enough together but if anything it would destroy one tank and the others would be knocked out. To me this is nothing more than German propaganda.
Articles:
http://www.achtungpanzer.com/sturmtiger-sturmpanzer-vi.htm
http://www.militaryfactory.com/armor/detail.asp?armor_id=404

Firing:
Inside the tank video:

It does. Take the turret off and you get a tank that looks like that. I wonder if that take was loosely based off of that tiger.

Chode gun

Nice post, glad to see others posting World War 2 stuff!