Is it $39/month and we can stream as many games as occur in that month? Or do we have to pay for each game we stream?

The cost of the service is monthly not by game, so for $39 per month, you have access to one stream that can be used for up to 50 hours per month for as many games as you need.

If you need access to more than one live stream at a time, you also have the option to add additional camera streams to your plan in your Account Settings.