It’s actually a little more nefarious than “the US had the biggest economy“. The US literally conquered, militarily and diplomatically, almost the whole planet during World War II, leaving only the Soviet sphere in Eastern Europe out of the dollar zone.
Since that time, the international rules — the institutions — for trade, diplomacy, and economic relations — have been written and implemented under the US custodianship.
That is how the US dollar became the reserve currency of the world, and that is why it continues to be so today and for the foreseeable future. Of all the currencies of the world, the one most likely to be exchangeable for what you want, anywhere in the world, remains the US dollar.
The World War I and II
The Bretton Woods Agreement
Finally, during the WWII, when there was clear who the winners are, the US proposed its currency to become the world currency (global reserve currency status) and assured the delegations from the 44 countries that the USD will be convertible into gold ($35 per ounce).
This happened during the Bretton Woods agreement, in 1944.
The world had been on the Bretton Woods system since the end of the Gold standard. That involved each country adopting monetary policy to maintain a fixed exchange rate by tying its currency to gold. They had the right to convert their currency notes for gold which we had on reserve. eventually the global economy outgrew our supply of gold.
“Nixon shock was a series of economic measures undertaken by United States President Richard Nixon in 1971, the most significant of which was the unilateral cancellation of the direct convertibility of the United States dollar to gold.”
When the world powers met together at Bretton Woods, they were trying to put together a lasting system to keep the world economy humming along.
One of the members of America’s delegation was Harry Dexter White, who outmaneuvered the rest of the negotiators, and most specifically, the UK delegation, to get the US dollar set as the currency against which all others would be measured.
During the bulk of the negotiations, White planted the idea that all of the currencies would be tied to some metric to be determined in the future. With all of the big plans figured out, White somewhat surreptitiously passed an amendment that provided that the to-be-determined-in-the-f
In short, it had nothing/very little to do with trust or belief in the US Dollar (as much as Americans like to think that’s the explanation). In fact, the United Kingdom was very opposed to the US Dollar being the reserve currency for the world.