The yeast Saccharomyces cerevesiae is a very important agent in the world of food and beverages. Its abilities have been exploited in the production of foods since ancient times before microbiology was known. Louis Pasteur was the first to demonstrate in the mid 19th century, the biological action of yeasts on sugars in the absence of air (fermentation).

Scanning Electron Micrograph of Saccharomyces cerevesiae, the yeast which brews

In baking, it is a leavening agent where it metabolizes fermentable sugars in the dough or cake mix, to produce carbon dioxide gas. This gas forms pockets /bubbles in the bakery preparation and gives it a soft and spongy texture which is fixed when baked.

In beverages, the yeast metabolizes sugars in the brew wort to produce both ethanol and carbon dioxide. This is what gives beer and champagne both its kick (alcohol) and frothy/bubbly nature (gas).

In root beer, the ability of yeast to make gas is what is captured rather than its alcohol-making quality. Although called a beer, if properly made, root beer is not noticeably alcoholic (less than the 3 - 5% of most beers). My guess is that root beer has the latter noun because it froths when poured, just like a beer i.e. it looks like a beer.

This is how I made a batch of root beer.
 

• You are making a food product.
  Exercise maximum hygiene/sanitation in all your procedures.

• There is the possibility that the bottles used for root beer production may explode under excess stress created by the carbon dioxide released during fermentation. Choose stout bottles.

Prepare the root/sugar substrate

• Add 280 g of brown sugar (you can make up some of this weight with molasses sugar to give more flavor and a darker color) to 2 litres of water.
  Brown sugar has micronutrients which aids yeast propagation during the brew.

• Add 3 tablespoons of McCormick Root Beer Concentrate (or similar)

• Bring the mix to the boil in a saucepan with a lid and keep it there for at least 15 minutes to ensure that the mix is reasonably free from microbes which might compete with the yeast for the sugar.

• While the mix is still hot, withdraw
  One aliquot of 10 millilitres into a hot water washed cup, and
  Another aliquot of 190 millilitres into another hot water washed cup.
  Cover both cups with a lid (you can use aluminium foil).
  Close the lid of the saucepan containing the rest of the sugar syrup tightly and set aside in a cool place (refrigerator is good) until the yeast inoculum is ready to be added.

• Let the sugar syrup in the cups cool down to room temperature.

Prepare the yeast inoculum

• To the cup with 10 millilitres of sugar syrup, add a few grams of
  "Mauripan" instant dry yeast (you can use any other brand of yeast).
  Note that Mauripan is a product selected for baking, but this yeast strain will make brews just as well.

• Let the yeast activate over some 6 hours.
• Pour the contents of the first cup (10 millilitres) into the second cup and let it sit for some 6  - 12 hours.
• When you have enough live yeast in the cup it should have a yeasty (bakery) smell and you should see bubbling and froth.
  This is the yeast inoculum which you have created by building up yeast cell numbers from a dry, inert state to a culture which is active.
• Note that if you have followed all the previous steps and an active yeast culture does not result, then the batch of instant dry yeast has probably expired and no longer capable of giving rise to live yeast cells.

Pitching the sugar syrup

• Pitch the entire 20 millilitres of yeast inoculum into the main sugar syrup which you have set aside.
  You should now have about 2 litres of starting brew in which fermentation of the sugars will start immediately.
• Decant your pitched brew mix into bottles with tight sealing caps.
  I used 500 mL IKEA "Korken" bottles.

The root beer brew just after decanting into bottles
 

Fermentation

• Set aside in a warm place for about three days (or more) to let the bottles brew.
  You should test the bottles each day to check for the amount of carbonation which has developed.
  You may stop the fermentation at any time which you feel that enough gas has built up.
  Stop the fermentation by placing the bottles in a refrigerator. The cold will hibernate the yeasts' metabolism.

Enjoy your bubbly root beer!