PracticallyEdible has a nice description of Devil's Food Cake.
Originally, Devil's Food Cake had a medium dense texture. The colour had a reddish tint that was probably caused by baking soda reacting with cocoa powder. In fact, I have an old cookbook (The Day by Day Cook Book, 1939) that contains a recipe for Red Devil's Food Cake. This recipe calls for 2 oz. unsweetened chocolate and 1 tsp. of baking soda.
There are lots of people who have a rather simplistic approach to nutrition and think that removing fat and calories makes you healthy. Then they go through recipes for things they want to eat, replace the sources of fat with something which doesn't have fat and doesn't make the result outright inedible, and declare their recipe a success. I think this is what happened here.
In a cake, eggs provide leavening, moisture, smoothness, own flavor, and enhancement of other flavors. Oil provides smoothness and enhancement of other flavors (and possibly its own flavor, if not netural). And while it is not water based, it keeps the moisture in the cake from evaporating, so it makes the cake less dry.
If you are a "simplistic nutritionist" without all this information, you can approximate some of the effects with soda. It will provide moisture, and it will also provide some leavening because it is fizzy. It will provide some flavor of its own too, but frankly, I find the rather chemical flavor of soda to be unpleasant. And it won't have any fat. In the eyes of the simplistic nutritionist, it has successfully replaced the oil and eggs while reducing fat and calories.
From the point of view of a baker, the cake will be a disaster, and won't even deserve the label cake. It will dry out quickly because it has no fat. It will have a bland flavor. Its texture will be terrible. They say "more chewy?" It will miss both the protein structure and the emulsifying agents provided by the eggs. It will be essentially an overwhelmingly sweet quickbread with no redeeming qualities. From a culinary point of view, it will be terrible.
Bottom line: under some assumptions, it is a good substitution. For me, these assumptions are so far from reality as to be useless. It is a terrible substitution.
Best Answer
For this recipe, the most likely issues are:
overbaking and wrong oven temperature
Simply put, removing too much humidity during the baking process. Wrong oven temperature can aggravate this problem, especially if it is too low (to hot = burned edges and wet center).
over-mixing
This recipe is very sensitive to overmixing, which means forming gluten strands that make your cake dense. Alton Brown specifies the mixing time in seconds for a reason. As a beginner, you could use a whisk instead of a mixer and stir just until the "just combined" or "no more lumps" stage, not more.
Technically it uses the same technique as muffins, not the beat eggs / butter / sugar until fluffy technique known from cupcakes
waiting before baking
This recipe gets its "lift" almost exclusively from baking powder. It is activated the moment it gets wet and then even more when heated. So if you don't bake the batter right away, it may "fizz out" somewhat, leaving you with a dense cake. This will take longer to fully bake and again be dry.
wrong measurements
The recipe states "ounces", which is a weight unit. I other words, you need a scale to measure your flour and cocoa. While a (US) fluid ounce is two tablespoons or 30 ml, an ounce is 28.3 grams. So for water, you can roughly exchange one for the other, but never, ever for other materials, especially light and dry stuff like flour.
(Yes, I have seen this happen.)
From your comments, overbaking might be the main problem here, possibly combined with overmixing.