Earth had gone through some crazy stuff over the centuries of history between modern day and the time of the original Star Trek series. There were the Eugenics Wars (Khan was heavily involved with those) followed by a global civil war of sorts and then finally culminating in World War III, which resulted in genocide and nuclear holocaust. Something like 30% of the entire planet’s population was killed off. A decade after the war ended, Cochran developed the first prototype warp drive and Vulcans made contact with Earth, and began to share their technology and other teachings. In the aftermath of all of that stuff, it would be no surprise if there was some sort of homogenization of Earth culture. Clearly there are some linguistic changes, as we see accents (Scotty and Chekov in particular in the original show) but it appears that English was adopted as an official language. Remember too that Jean Luc Picard was French, but spoke English and had no French accent because by the time he was born the French language was no longer actively spoken.
In Star Trek: Enterprise, which was set long before the original series and represented Earth’s first attempt at deep space exploration (and was even set before the founding of the Federation), they had to bring a translator onto their ship (Hoshi Sato) to help them figure out how to communicate with other species. There was a translation computer (provided by the Vulcans), but it was not sufficient to properly translate between English and the language of every alien species they encountered, so she often had to use her own linguistic skills, and in particular she was stated to be the first human fluent in Klingon language. She was raised as a polyglot and could speak many Earth languages (and was a language teacher before her time on the ship) but I got the impression that languages were learned more for historical reasons, and less because it was an actual necessity on Earth by that point.
So if Earth had pretty much formed one nation and had one language, then it made sense that most other planets were like that as well.