What have been the drivers of the globalization of politics?

While there are several ways to answer this question, the biggest reason is that it's in the best interest of countries to work together. Some scholars of International Relations believe that countries do what's in their best interest (just as people often do).
When countries work together and create international political systems, such as the United Nations or the European Union, it helps the world advance together in sectors like technology, trade, and transportation.
The globalized political system is also closely tied to the global economic system. Companies are able to cross borders and earn money like never before, which helps to boost the economies of the countries involved.
Let's take a look at two specific examples. The United Nations was primarily created to minimize conflicts between countries and to put a set of rules in place when it comes to war and international norms. The European Union was created to make things easier for European countries in terms of trading, transportation, immigration, and so on.
Long story short, there are a variety of factors that drive international politics, but the two largest are likely economics and security.
https://history.state.gov/milestones/1937-1945/un