The reason that an automatic doesn't stall out while "in gear" and at a stop, while a manual transmission does, is that automatic transmissions use a hydraulic torque converter to connect the engine to the transmission, while manual transmissions use a friction clutch. These two systems do a similar job in a very different way.
A torque converter uses fluid to transfer the power, and so it can "slip," effectively disengaging the engine from the drive wheels at low speeds. When you come to a complete stop in an automatic transmission car, the torque converter starts slipping, allowing the engine to keep turning even though the wheels have stopped.
The only way to get the friction clutch of a manual transmission to slip is to step ondepress the clutch pedal. If you come to a stop in a manual transmission car without depressing the clutch pedal, the engine stops turning when the wheels stop turning, and the car stalls.
See wikipedia'sWikipedia's articles on torque converters and clutches for a more thorough treatment.