The quantity of which ingredients should be increased to raise the moistness of a cake?

Talking about the role of each single ingredient and how they interact is way too much for an answer here, and parts of it have been discussed in other questions anyway. So I will give a quick info on the recipe you linked:

It has way too much flour. For a standard pound cake, you want equal weights of flour, sugar, fat and egg, and most muffins follow a similar ratio. Some of them add liquid. But what you have here is a recipe which adds 50 g of sugar, 55 g of egg and 60 g of oil to 130 g of flour. Luckily, the oats are soaked, so they are probably not sucking out additional moisture from the batter. But still, you should change the ratio to get a standard muffin. Or, better still, you can get a recipe that works instead of fiddling with a bad one; finding a new, good recipe is generally easier than repairing.

There are a few possibilities for why this recipe turned out dry.

First you could have over cooked the muffins. They should be springy but also tacky and still moist when coming out of the oven. Baked goods will continue to bake after being removed from the oven, so its best to take them out when they are a little on the "too soft" side, as they will firm up and have a perfect texture once left to rest. If they are taken out too late, they can end up being dry.

Another possibility is that the flour and oil was over mixed. You don't want to develope the proteins (gluten) in the flour, as that will make the muffins dry and tough. You will want to mix until it just comes together and then stop. It is easier to prevent gluten development if you use pastry flour instead of all purpose or bread flour. I find pastry flour gives a better texture to short breads such as muffins.

If you don't want to use a pastry flour, you can replace some of the flour in the recipe with corn starch or rice flour. This change will increase the starch content and decrease the proteins which produces a moist crumb.

If those changes don't help, there are a few more things you can try. Replace the white sugar with brown sugar, or increase the sugar content a little bit so the end result is more moist. You can also add fruit puree like applesauce or mashed banana to increase the moisture of the finished muffins. If you don't mind increasing the cholesterol and fat, you can add an extra egg yolk to the mix.