That would be your Torque Converter. There's a impeller, stator, and turbine inside. You can look up how it all works on the how it works website.and started rattling off something about torque and a stater (stator?) and something about not locking and backspinning.
BUT... basically if that is shot, you more than likely don't need a whole new transmission, especially if you're driving down the road. And if transimission was "spraying" fluid all over the back of the engine, that sounds more like a broken line than just the torque converter locked up.
I would definitely get a second opinion on what is wrong. $4200 is a lot of money to fork up, especially if you don't need the whole transmission. You've already got the car out of the "stealership" so you're at a better position already.
The ECM reprogram is a service that will be covered by Toyota, and should not cost you a dime.
(P.S. for those that might get offended at my "stealership" comment, I am a ASE certified mechanic in both areas of Light and Heavy Duty vehicles and spent 2 years working on automatic transmissions. I'm no way the "absolute" expert on them, but I can tell you something smells fishy. A good service department will explain everything that is wrong)