Unfortunately, my sources for this are in Swedish and/or from old periodicals from the actual time when Esperanto was new, so I can't link anything. If you can read Swedish, find the book called "100 years of Esperanto", and in Esperanto there's another book you can try, "La Esperanto-Movado en Upsalo" (as I remember the title).
Basically what happened is, the book first came out in Russian, and from there people just found it as something amusing or interesting to do in their free time. That's how it spread very minorly to other countries, through bilingual people who translated it. Once it got to Germany, some people who had visited Germany and who were students at the University of Uppsala noticed it and decided to not only form a club (there was also a German club at this time, but it was the only club in the world I think) but also, when they heard that Esperanto was starting to be banned in Russia/a Russian Esperanto periodical was closed down, they started their own which grew to be very big and influental in the first few beginning years. (Swedes are easily outraged about injusticies so honestly that's probably why they started it, also Uppsala is very well-known for being a "linguistic research capital".)
They managed to get funding from people outside of Sweden, among other things, and used it to distribute flyers (they also got flyers from Zamenhof himself and wrote to him a lot, asking about stuff). Their periodical was simply called "Lingva Internacia" or something like that, but it included all sorts of stuff, like both local Swedish news and Esperanto news. The club and so on was written about in the Swedish newspapers too, among other things, so that's how it got some more publicity. Basically for about five years, Uppsala was the "capital of the Esperanto world" due to this, and they influenced other groups to start making their own periodicals. I believe the very initial list of potential subscribers was obtained by mailing Zamenhof himself and asking about it.
When other countries started having more periodicals and clubs too, they also started being advertised in the countries' local newspapers - this was sometimes in a normal news-y way, but other times the members actually paid for advertising in the hopes that it would make new members; this was a very expensive thing to do at the time. This doesn't mean that the clubs or periodicals were any big (it only takes two people to form a club!). Don't forget that this is also a time when you did normal advertising in the newspaper, so anyone looking for an Esperanto penpal or teacher would likely use the newspaper to place the ad. It was also possible to subscribe to periodicals even if you lived abroad, so ex. there were French readers of the English Esperanto periodical. Of course, this was also a different time from today, where it was extremely normal to try out new things and different clubs and so on.
It seems like most of the people simply heard about the language from friends or from various newspaper articles about it. A lot of people simply told even strangers they met about it (ex. on the train). Even back in the 1920's and earlier though, there were elementary schools (among other things) who were teaching the language to children in preparation of either "the general future" or for them learning another language later. The craze about this was honestly slowly growing to be very big, even train personnel and the police in a few different countries were being instructed in Esperanto for training for dealing with foreigners. Though of course, the world wars and then the monolingual English speakers growing to power really squashed a lot of it. And of course, most people used it as a hobby instead of thinking about it as a serious world language, but they did at least consider it fun so that's how it was propelled.
I also remember reading that news about Esperanto was randomly broadcast on the radio in its early days; that's how I heard it reached the Faroe Islands for example. Also, a lot of the people who initially learnt the langauge were very educated to begin with (ex. rich French people, rich Swedish University students), so they had some influence on other people already.
Those days were also different from today in that people needed a world language, but there really was none - English certainly wasn't big (German and French were much bigger!), but it was an age of new technology, new ideas about psychology, and easier/cheaper travel between countries and so on. But honestly I think the main point was that people were bored and thought learning a new language was simply fun when they heard about it, and it was an easy way to make foreign friends and get foreign news (this is a point I've read about quite a few times when I read texts from the same era - the local news was usually not at all accurate regarding foreign countries, and people loved collecting stuff like stamps and postcards).
Edit: Something I forgot! I've read that people used to take a text (ex. a book) in Esperanto, and then translate it in their head, reading aloud, in real-time in front of their friends in order to convince them of how easy it was to understand and how fast it was to learn. This is mentioned in that one book that's a collection of Zamenhof's answers about grammar and vocabulary questions.
Teachers were mainly pressing the fact that it was a good test of how good students were at language ("if they fail at Esperanto, we know they're no good at language and it's useless to teach them ex. French in secondary school"), and that it really gave the kids enthusiasm about history and geography, since the teachers were having the main part of learning Esperanto center around having the students send letters to lots of different countries in it. This is mentioned in quite a few places and pretty much corresponds with what modern educational tests with Esperanto are saying today, but I remember one source for this is here: