In order for planes to fly, they have to modify air pressure. When air pressure drops suddenly, water turns from gas to vapor. It's really cold up there so it freezes and remains as a trail.

If you fully understand the physics behind flight, then you will understand the vapor trails.

If governments are so sophisticated that they can modify weather, wouldn't they makes these trails less visible?

Kaku is just saying that if you heat up the water vapor with a laser, it can't condense into clouds over a local area.

If you don't believe this, get a syringe and half fill it with water. Remove the air and put your finger over the end. Pull back the plunger to create a vacuum. You will notice water vapor. That's basically what the wings are doing as they pass through the air.

Do you think I work for the government?