There is certainly more to the story, and I write this at risk of accidentally writing a lecture about Esperanto and other conlangs, but they do play into the story: when the League of Nations was formed, Esperantists did try to make Esperanto an official language recognized and utilized by the League, and this measure was discussed, but it ultimately failed.

For a long time, humans have dreamed of there being a universal language, one that everyone can speak. (In the 1993 play "The Universal Language" by David Ives, a character says “I don’t think language is just music. I believe language is the opposite of loneliness. And if everybody in the world spoke the same language, who would ever be lonely?” Which I think is a pretty neat sentiment.) There is of course the legend of Babel: in Genesis, following the Great Flood, all humans spoke the same language, and tried to build a tower to Heaven; seeing this, God destroyed the Tower, split up the people into different nations, and they all spoke different languages (Genesis Chapter 11). Since then, there have been people hoping to find a way to restore the unity of humanity, and some people have tried to do that by creating new languages. Particularly in the 19th century, there was a trend of constructing new international auxiliary languages (or auxlangs), languages designed to be secondary languages that are accessible to people regardless of their native language, so that people of wildly different cultures that don't share a primary language still have a common language they can communicate with. Until the 1880s/90s, the most popular auxlang was Volapük, a language designed by Johann Martin Schleyer, but a schism developed in the Volapük community and it lost popularity.

In the 1887, Ludwig Zamenhof published Unua Libro ("First Book"), the first book on the grammar of Esperanto. The book was also titled Dr. Esperanto's International Language; Zamenhof called himself Dr. Esperanto because the word literally means "the one who hopes", and eventually that became the name of the language (in Esperanto it would be referred to as la lingvo internacional until Esperanto caught on). Zamenhof sought to make a language as easy to learn as possible, regardless of your native language, so he gave Esperanto vocabulary and grammar rules familiar to many different kinds of languages. The philosophy of Esperanto, both as a language and a movement, had three components:

1. To render the study of the language so easy as to make its acquisition mere play to the learner.

2. To enable the learner to make direct use of his knowledge with persons of any nationality, whether the language be universally accepted or not; in other words, the language is to be directly a means of international communication.

3. To find some means of overcoming the natural indifference of mankind, and disposing them, in the quickest manner possible, and en masse, to learn and use the proposed language as a living one, and not only in last extremities, and with the key at hand.

Again, at risk of this becoming a lecture about the history of Esperanto when it shouldn't be, I'll try to summarize the next couple decades in a few quick sentences. Esperanto is published in 1887, and it develops a decent following, bigger than most conlangs had acquired (to this day, Esperanto is one of the most successful conlangs, the only competition being languages used in fictional works like Lord of the Rings and Star Trek). Various groups, such as the Sennacieca Asocio Tutmonda (World Anational Association), tried to use Esperanto promote international communication and relations, but—much like most Esperanto efforts—were not as successful as they'd hoped. Esperantists developed finavenkismo, the belief in the "final victory" where Esperanto succeeds at becoming the lingua franca of the world, and everyone is speaking it as a secondary language. There were two methods of achieving this, as Zamenhof noted in a 1910 speech: privately promoting self-study, or getting governments on board; Esperantists tried both.

World War I happens shortly after Zamenhof's death, and following its conclusion, the League of Nations was founded in 1920. And the League did discuss Esperanto in some capacity; Inazo Nitobe, Japan's undersecretary for the League, noted that Esperanto could be good "for meeting the demands of science and commerce and the still higher needs of an instrument for higher understanding." The League didn't decide to make Esperanto an international language, or an official one used for their purposes (their official language became French and English), but they did consider encouraging Esperanto study. They sent out surveys to member countries, and found relatively decent activity in countries across Eurasia. Countries were on board to support Esperanto education in schools. However, fears of anti-nationalism and communism led to countries trying to shut it down internally, and this lead to efforts that would further bad-mouth Esperanto, shutting down the effort—and in turn, finavenkismo—for good in August 1923.

I'm not sure what happened with the League after that, or the United Nations later on when it replaced the league in the '40s. I will note that Esperanto was not the last attempt at creating an auxlang: the International Auxiliary Language Association was founded in 1924, and published Interlingua in 1951. Interlingua is designed to be easily understandable by speakers of any Romance language, but it it has never reached the popularity or impact of Esperanto.

I know this doesn't give you the whole picture, but I hope it gives some insight into the goals of a universal language.

Main Sources:

Okrent, Arika. In the Land of Invented Languages: Esperanto Rock Stars, Klingon Poets, Loglan Lovers, and the Mad Dreamers Who Tried to Build a Perfect Language. Spiegel & Grau, 2009.

Schor, Esther H. Bridge of Words: Esperanto and the Dream of a Universal Language. Metropolitan Books, 2016.

Zamenhof, Ludwig. Dr. Esperanto's International Language. Edited by Gene Keys. Translated by Richard H. Geoghegan, Oxford, 1889, https://en.wikisource.org/wiki/Dr._Esperanto's_International_Language.

Post WWI, there was indeed a push for one universal language to be used by the whole world.

It was hoped that by having the whole world able to use one language to communicate, it would stop the atrocities that plagued the world in the prior conflict.

There has always been an idea of a universal language, yes, these are called Lingua Francas, and they date back to Ancient Greece, where surprise, surprise, Ancient Greek was the go to language for anyone who was anyone. We then had Latin, thanks to the Romans, which is where we get the term 'Lingua Franca' from, meaning 'common language'.

Now, or at least in the past 150 years or so, linguists have agreed that no natural language deserves the right to be made the universal language, no matter how much the British, French, Spanish and Americans all hoped it would become.

Most contenders for the universal language have been construct languages such as Edo or Esperanto.

After WWII, most people had some sort of American or British garrison just a few hundred miles from their shores at any one time, so it was generally considered that English was to be the main Universal Language, which led to the decline of both Edo and Esperanto. Which is a shame really, because Esperanto is such a beautiful language at heart.