help understanding this recipe gluten free and vegan

You have several oily ingredients like coconut oil and nut flour. Oil is one of the 2 ingredients that make a cake moist and it helps retain the moisture. The other is simply water.

In addition to plain water there is xantham gum in the recipe. Xantham gum binds a lot of water and retains it, keeping the cake from drying out. That's why only 1g of it is needed to incorporate 100g of water into the cake. If you leave the xantham gum out of the dough, the cake would probably be very soggy if it wasn't baked long enough or too dry if it was baked for too long. Hitting the perfect spot between soggy and dry would be much harder and the cake would dry out within a day or two after baking.

And lastly there is rice flour and sweet potato. Both of them contain starch (and the sweet potato also adds a little more water to the cake). When starch gets cooked (or baked in this case), the molecules swell and can retain moisture much better than uncooked starch. You can see the effect when cooking a pudding / flan / blancmange.

You seem to be under the impression that only potato starch can replace the wheat starch provided by wheat flour in standard recipes. This is a misconception - any starch will do, and besides, the more other carbohydrates you have (sugar, fibre), the less starch you need, as they also store moisture and are to some extent interchangeable in this function. The other ingredient that gives cakes a moist mouthfeel is fat.

So, to look at your ingredients:

• nut flour: has starches, fibre and fat
• rice flour: that's pure starch
• 1g xantham gum: a fibre, but not directly connected to moisture (it is used for the leavening to work)
• water: beside liquid-retaining ingredients like carbohydrates, you also need the liquid
• coconut sugar: sugar
• coconut oil: fat
• cacao: starch and fat
• salt: doesn't contribute to moisture
• cake yeast: doesn't contribute to moisture
• vinegar: same as the water
• sweet potato: not only is it full of starch, sugar and fibre, the fibre is in the form of already-hydrated hemicellulose and similar, which is excellent for a moist texture in baked goods.

So, at the end of it, most of your ingredients work the way you assumed potato starch should. In fact, some of them are also used in wheat-based recipes in order to make the cake moister, since they work better than pure starch.