How do you significantly reduce the calories in bread like Franz Keto bread?

A comparable keto bread recipe is Diedre's Ultimate Keto Bread 2.0. It uses a combination of golden flaxseed meal, oat fiber, and vital wheat gluten as the flour ingredients. I've made that recipe probably over 100 times while making different tweaks to improve on it. I've arrived at a couple different variations, and there are some tweaks that I'd recommend if you'd like to lower the calorie count.

86 g golden flaxseed meal         26.38%     86.0 g (flour)
86 g oat fiber                    26.38%     86.0 g (flour)
154 g vital wheat gluten          47.24%    154.0 g (flour)
                                 100.00%    326.0 g (total)
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7 g Kosher salt                    2.15%      7.0 g
16 g instant dry yeast             4.91%     16.0 g
8 g honey                          2.45%      8.0 g
280 g water                       85.89%    280.0 g

Sorry, the recipe is in grams and I don't have volumetric equivalents. The ratio of flour ingredients to each other is important, and the only reliable way to measure them is to weigh them. These are bakers percents. The top 3 ingredients are considered the flour ingredients that have a total weight of 326 grams. The percentages listed for the flour ingredients represent the proportion that the particular flour ingredient makes up in the total flour; their percentages will add up to 100%. The percents listed for the last four ingredients are based on how the weight of the particular ingredient compares to the total weight of the flour. For example, 7 g of salt divided by 326 g of flour equals .0215 or 2.15% of the weight of the flour. Bakers percentages are extremely helpful in trying to understand how non-standard recipes work. For instance, you'll always need the vital wheat gluten percent to be around 45% to 55% for it to be able to properly make bread.

Basically, I'd definitely recommend removing the xanthan gum from her recipe, as it inhibits gluten development and results in a tighter crumb. The butter in her recipe adds taste, but can be left out if calories are a concern. The 2 eggs she uses aren't strictly necessary but they may help strengthen and open up the crumb somewhat, while also helping as a drying agent (flaxseed meal and vital wheat gluten are very hygroscopic). In lieu of eggs I'll sometimes add 5% or so of resistant wheat starch or resistant corn starch though it may not be necessary. The honey serves as a source of food for the yeast, since none of the other ingredients are fermentable. Most of the honey will be consumed by the yeast so it doesn't add much to the overall carb count.

While she uses a stand mixer in her video I do want to add that it is possible to knead this by hand, though there is a bit of skill needed. Thanks to the vital wheat gluten, the dough comes together quite quickly, though it starts off somewhat slimy and sticky (from the mucilage in the flaxseed meal). Use a dough scraper to repeatedly fold it over on itself and knead into the counter. Eventually it may start to dry up and you can start kneading it by hand. It may take 10 - 15 minutes worth of kneading to develop and distribute the gluten evenly into a network that will trap the gasses of fermentation. You should be able to smear and stretch the doughball with the dough knife/scraper against the countertop and it shouldn't tear while doing so.

Unlike traditional bread, which uses a bulk fermentation and then another rise in the pan, you're better off doing a single fermentation in the bread pan (since there's not a whole lot of food). If the loaf ends up dense, it likely wasn't kneaded sufficiently.

How this recipe works: vital wheat gluten mixed with water makes a virtually solid rubbery mass. Oat fiber is a pure insoluble fiber that interferes with the gluten networks such that it prevents cross-linking and allows for air holes to develop during fermentation and baking. Flaxseed meal contains soluble fiber in the form of a mucilage network that can be woven in between the gluten network to help open up the crumb somewhat.

As far as calorie count: the recipe given is about the lowest I've been able to make it without taste suffering. Oat fiber is pure insoluble fiber meaning it has no calories, but it also tastes like cardboard (with the slightest hint of Cheerios). You can try raising it to a higher level to stretch the ingredients but taste tends to suffer. Flaxseed meal is very calorie dense: 28 g has 150 calories (from the fat). However, it's the only ingredient I've found that helps inhibit gluten development in a beneficial way (to open up the crumb).

As for what they are doing, you already put your finger on it - it is the fiber. Instead of making the bread out of flour, they are basically making it out of water, and adding sufficient fiber that they get a workable texture instead of a slurry.

For doing it at home, the most workable way is to search for similar recipes. Mixing some fiber with water is not difficult, but baking it and getting the result to look, feel and taste like real bread is a complex feat of food technology. There is certainly not one trick that you can apply and get the desired result, but as in any complex process, everything you do will have to fit everything else. Even if you intend to develop a new recipe on your own, it is highly recommended that you first try out baking several recipes of this type (and getting really good at them) before you attempt to evolve them further. So look around the Internet, or get a suitable cookbook, choose your recipes, and start baking.