It is quite easy to find the ingredients of the bread from the picture you posted. They are:
Purified Water, Organic Coconut Flour, Egg Whites, Psyllium, Organic Apple Cider Vinegar, Baking Soda.
J.A.I.L.'s answer already explains part of what makes bread texture what it is: you need raw proteins which form a flexible 3-d mesh to trap the air when they set during baking. You also need a thickener combined with water which makes the whole thing soft instead of rubbery. In wheat bread, the proteins are called gluten (actually a mixture of two different types of protein, both coming from the wheat), and the thickener is starch (also coming from the wheat). In the bread you posted, the protein is the egg white, and the thickener is the psyllum mucilage. Coconut flour doesn't even come into play; it consists mostly of fat and some fiber, but the fiber is not of the rare thickening kind (as opposed to psyllum fiber). It contains nothing helpful for the bread structure, so they obviously use it as a filler. Of course, it will provide you a pleasant mouthfeel, aroma and lots of calories; it just won't give you a bread-like structure.
Conclusion: Trying to build a bread recipe around any nut flour is totally wrong. You have to search for a recipe which has the right balance of protein source, thickening agent and leavening agent. Once you have found it, you can substitute your favorite nut flour (or some other food item) as a filler. By the way, making such recipes from scratch is very hard, because you need not only the combination of ingredients (in this example, it is easily found), but also the correct ratio, and the correct timing for mixing and baking; these artificially-constructed combinations are nowhere as forgiving as wheat doughs when it comes to leavening. I would suggest looking for existing recipes. Sadly, it would be off-topic to give you one here, even if I had one, but I can give you a tip for your search: stay away from the ones which don't have all three things: leavening agent, protein and thickener.
Probable protein sources are: eggs, vital wheat gluten, possibly extracted soy protein (if you can get a source for its raw form). Probable thickeners are: anything described in this e-book. As for leavening mechanisms, the Wikipedia article should give you enough pointers. If you find a recipe which has one of each, it is likely to end up puffy. I must warn you that alternative thickeners have a mouthfeel which is very different from starch, so most breads made from such recipes will probably be much less soft/cottony than white wheat bread, and some of them might have a slightly rubbery or gooey feel. But the good ones will have a decent approximation of bread texture.