The purpose of eggs in a cake mix is to help hold the cake together when it cooks and keep it moist so by adding additional eggs to the cake mix you create a stronger cake and also help make the cake more moist.
So the end result of adding more eggs to cakes is you have a stronger cake and a more moist cake.
So yes more eggs do make cakes more moist.
You can also use sour cream instead of eggs in cake mix which helps to hold the cake together when it bakes.
When you don't use eggs or sour cream or other dairy products the cake can fall apart more easily.
But you can make cakes without sour cream or eggs but it just won't hold together as well.
When baking a cake using already made cake mix in a box you add at least 3 eggs as well as some vegetable oil and then some water and mix the cake mix up using either an electric mixer or hand mixer.
I prefer to use an electric mixer to mix the cake mix up.
If you want to add more than 3 eggs you can do so but 3 eggs is minimum to make the cake perfect.
However if you cannot have eggs you can make the cake without eggs if you want too or need too.
You can use some applesauce in place of the eggs if you want too but I've made cakes without any eggs or applesauce and it turned out okay for me.
You can use milk instead of water as well to help with the taste of the cake when you don't use eggs.