Imagine you are a farmer, and all your friends are farmers, and all of your family are farmers. Some may specialize in certain kinds of farming, but that is what you do. One day, some guys covered in mud and blood emerge from the woods. You have a few options, but none of them good. You fight, and ultimately lose, your land is stripped bare and half of your village is killed. You sucumb to their demands of your food clothes and women, and this becomes an annual occurrence as the mob follows animal migrations. The final option is to be inducted into the tribe. All three options would account for aryan dominance through migration. You either die or are bred out.
Aryan invasion theory at least partly explains human migration patterns. Aryans over the history of mankind have migrated to different parts of the world. The country where their remnants are the most obvious is India. In India, the light-skinned citizens have a higher status than darker-skinned, but more so, they are considered to have the blood of actual Aryans in them.
I believe Aryan invasion theories do justify some of the migration patterns we see during their time. I think a lot of different forces can go into migration, so they may not be the whole story, but at minimum, they are a major factor. I don't think their impact was everlasting.
The Aryan invasion hypothesis was very widely taught in Europe for decades due to the idea of eurocentricism and modern European society being a superior race descended from superior people. Archaeological efforts shows native Dorian and Ionian Greeks living uninterrupted in their land for thousands of years - an invasion is not supported.
No, the Aryan invasion theory does not explain human migration patterns, because it is not a valid theory. People have always wanted to go explore new places. This is true for many types of races. It is true that white people were part of this exploration, but this does not make them invaders. They were explorers. They wanted new trade routes. They are no different than many other explorers who settled in places other than their homelands.