Nike internationalized rapidly largely because globalization in general was taking off at about the same time that Nike was.
Nike was helped by globalization in at least two major ways. First, global transport was becoming cheaper. Containerization was growing exponentially in the 1980s. This helped make it easier to manufacture products in Asia and ship them back to the US for sale. Second, this was a time when a boom in telecommunications was starting. Cable TV was becoming ubiquitous. Later, the internet came into being. These factors helped give American professional sports a worldwide audience. Soon, the NBA was big news around the world. As this happened, Nike was able to establish itself as a prominent brand around the world.
Thus, factors like better transport and better communications led to globalization. This globalization allowed Nike to find cheaper manufacturing and broader markets in which to sell.