Part of our perception of this is just formed by the archaic word choice, but mostly it is because people simply tried harder to write eloquently and weren't just putting everyday speech to the page every time they wrote. Although there are plenty of examples of letters from the period that sound perfectly normal and conversational, there was also definitely a culture of practiced letter-writing seen in etiquette books of the period.

In the introduction to Chesterfield's Art of Letter Writing Simplified from 1857 he describes the typical writing process at the time, [First the prospective writer tries coming up with a few openings of their own, but are not satisfied with their trite attempts]. 'By and by the writer begins to suspect that such a mode of beginning a letter is not very elegant and eighteen cents are expended on that very remarkable work "The Lady's and Gentleman's Complete Letter Writer, 90th edition." The time comes for another letter; the "Complete Letter Writer" is dragged out from the darkness of a drawer in which it had hoped to conceal itself forever, and an hour is spent in the search for a model letter that will just express the writer's feelings and ideas.'

Take a look through the mentioned works like The New and Complete Letter Writer, this edition from 1832. As he said, it just has hundreds of sample letters titled by occasion and the writer would search for a suitable one to model theirs on. Subjects as specific as From a young gentleman at a Boarding-school in the country, to his brother, an apprentice, in New York or From a Young Gentleman, in Expectation of an Estate from his penurious Uncle, to a Young Lady of small Fortune, desiring her to elope with him to New England (though that one is really just included for the young lady's response, just had to mention it though).

Similarly there were tons of popular books for snippets of poetry or literary quotes for people to peruse for use in correspondence, also generally organized by theme or sentiment of the poetical quotation. Editions of compilations like A Complete Dictionary of Poetical Quotations and similar were coming out annually in that era and were immensely popular.

He also suggests (somewhat humorously) that writing a letter is an activity that often takes an entire day for many people. So I think your answer is that mostly people just tried harder, often turning to sample books and putting lots of effort and time into their correspondence. That in addition to use of now-archaic vocabulary. People just aren't spending hours and pouring over books of poetical quotations and sample emails these days, which is ultimately just down to culture, not really education. Of course not every letter was written using these guides or samples, but their widespread popularity is very suggestive of the cultural expectations for letter quality at this time.