As per using amazon to scale, it depends on the services they use. EC2 is expensive, but you don't need that many ec2 instances to handle the streaming, outside of permission handling etc.

As for transcoding videos, they probably have a far more burst-like intake of movies, yet not a very large burst at that. How many tv shows are released per day? How many movies come out per day? Not too many, compared to a service like youtube that has to transcode huge amounts of information all the time. EC2 at least lets them turn on machines until they finish, at which point they can easily scale down after the burst is handled.