It's a high voltage capacitor. Use something with an insulated handle to short the terminals. Those can hold a charge for a long time unplugged. Much like the old picture tube TVs. Do not touch anything in there until you do this. The charge is enough to be fatal if you get across it. One rule to follow is to keep one hand away from the device. That way, any charge you get on doesn't have a path through your heart. He's using needlenose pliers to short them.
In reality, the danger is low if you treat it with a little respect. It's just not as obvious as a spinning lathe chuck that it can hurt you. In 90% of cases, there is a bleed resistor on those that discharges them over a few minutes, but those can fail.
Ebay has the transformers, even new ones, for about $20. That said, old microwaves are probably available free most anywhere. Older is better. Some of the newer ones use a switching power supply and don't have these big cores to harvest.