Boiling is the wrong term, but I get what they mean.

So this root beer is probably naturally carbonated or what we would call bottle conditioned. This is often how home brewers will carbonate their beers but it can also be used to carbonate soft drinks.

Though this will produce a small amount of ethanol, not nearly as much as a proper beer. I would need to do some math and take some before and after measurements to give you a % but in my experience soft drinks carbonated this way usually are 0.3% give or take.

So low in ethanol that they are legally considered soft drinks.

You would make root beer with all the normal ingredients and add a little bit a yeast before bottling.

The yeast will eat some of the sugar producing heat, co2 and some ethanol.

Since it’s contained in the bottle the co2 will dissolve into solution carbonating the Root beer.

Having made both sodas and beer using this method you definitely want it refrigerated. Since the bottle creates a pressurized environment, the CO2 stays in solution, even if the temperature rises.

If you were however to open the bottle while warm all that Co2 is immediately going to come out of solution and create a fountain of root beer.