Lets tackle the "how" first. One of the major structural changes of modern markets is the deterritorialization of money and the configuration of a global currency space. In the past, we had neatly confined maps of divided territorial currencies and those were only traded with outside parties in a very limited fashion. While not 100% accurate, you can view Currencies before the Second World War as insular, strictly controlled systems. Starting after the war however, these controlling barriers were slowly fading. Today usage of international currencies is no longer confined to the territory of the nation state it originates from. This was mainly caused by the intense growth of capital mobility in the last years, which coincides with the increased maturity of the Internet-Communication-Technology-Revolution, the increased volume of international trade, and the promotion of international free trade standards and practices. This deterritorilization of money is one the most noticeable effects of globalization. Today, at least one third of the world's money supply is now located outside its country of issue and global foreign-exchange markets have seen a massive surge in volume. Just to give you a small idea about the scale we are talking about here, from 1989 to 1998, the average daily turnover in according to the bank of international settlements in global exchange markets has increased from 590 billion USD to 1,5 trillion USD. This process has of course been going on for a long time and can not be accurately pin-point to a single starting date. But it has definitely been driven by intense market rivalry and free financial trade. As you point out yourself, some currencies come to prevail over others and are in higher demand on the global market than others. So lets take a look at what attributes are important to an currencies success.

The first and in my personal opinion most important attribute is the markets confidence in a moneys future value backed by political and economic stability in its country of origin. A singular currency becomes more attractive the more you can forecast its purchasing power in the future. Valuable benchmarks might be a long history of low inflation and the lack of great political upheavals. The market has to view the currency as a legitimate currency that will not lose its value in the future. In addition to that, a currency has to be convenient to exchange and have low transfer costs. To achieve this, the currency has to be traded on an open market, with low transaction costs that is not discriminating against foreign individuals or institutions. The barrier of entry has to be low and there need to by the necessary legal frameworks in place to facilitate a healthy market. Third you want a broad transactional network. The more other stakeholders are accepting and using a foreign currency, the more attractive it becomes. A large national economy is one of the most prominent kick-starter for such a development, but the true kicker is a leading economy that imports and exports huge volumes of products globally, as well as a political regime that supports the international use of its currency.

Together these three attributes determine which currency or currencies any stakeholder will want to acquire, both as a medium of exchange and to store his generated value. This is important, because the deterritorialization of money means that currency stand in direct competition to another. Being traded on global markets means that, just like you can expect from the product trade, a natural market hierarchy will occur, with some dominating market leaders and a lot of currencies that are unable to compete. The dollar is without any doubt the current king of global currencies. And the reasons for it success may now be rather obvious. The US Economy has been the leading national and international economy of the 20th century. The political climate has been stable and they did not have to suffer through a massive war on their on territory. The US has also spearheaded the international dogma of free trade and supported the creation of international markets. To further support this, the US has provided other states with massive infusions of USD and the USD has for a long time been one the most stable currencies of the world, backed by a massive economy that is expected to grow further and has been a leader in technology advancements for decades. The USD has never been devalued and inflation rates have consistently been low. This creates a self-reinforcing system. Because the USD is one of the most used currencies in the world, with 40%-60% of its total volume circulating in foreign countries and enjoys a great deal of legitimacy, there is a great incentive for stakeholders to use the USD as a stable currency.

When the USD became the "global currency" is not easy to identify. Before the USD, the Pound Sterling was without any doubt the reigning currency, since Britain not only ruled over a vast colonial empire, supported by a massive economy, but was also the leading banker of the world with over 4 Billion Pound foreign investments at the brink of WW1. Many economic historians agree that following the turmoil of WW1 and the Great Depression, the downfall of the Pound Sterling and its perceived legitimacy allowed the USD to take over the role as leading currency.

I will get back to your on the question about the impact of the deterritorialization of currencies on the sovereignty of other states later or tomorrow since I have to make dinner now and it is not a short or easy question to answer.

Sources:

Eichgreen, Barry, Arnaud, Mehl, Chitu, Livia, How Global Currencies Work: Past, Present, Future, 2017.

Kenen, Peter, The role of the dollar as an international currency, 1983.

Ganziro, Taranza, Vambery, Robert, et al, The Exorbitant Burden: The Impact of the U.S. Dollar's Reserve and Global Currency Status on the U.S. Twin-Deficits, 2016.

Cohen, J. Benjamin, The Future of Global Currency: The Euro versus the dollar, 2012.

Edited for formatting and to fix my mediocre english.