Why do people from the past appear to speak/write so formally?

Formal versus Informal writing.

You are sending a letter back to your honey? Nigga, that's formal.

You are carving graffiti into a wall? Nigga, that's informal.

Both are shown, we just have a shit load more pieces of formal writing. The why of it is precisely the same as now, except perhaps social standards have changed to allow for the conflation of informality into typically formal areas.

http://www.pompeiana.org/Resources/Ancient/Graffiti%20from%20Pompeii.htm

OP a crucial things to understand is that through the ages, what survived the most were official documents which have always been and still are written very formally. In 2000 years there are more chances that what will survuve will be UN resolution texts than some some random text messages written by some dumb teenagers.

I think a good example would be Cicero, because amazingly enough his private correspondence that was never meant to be published managed to survive to our days. There's a stark contrast between the official Cicero of "In Catillinam" where the text is clear and his private mail. In Catillinam is concise and very epurated, to a level it's considered basic translation material for highschool courses because of it's form (if you can't be bothered to learn latin to appreciate this, just listen to any Obama speech from 2008-9, they're all very Ciceronian).
On the other hand his private mail is a whole different story. It's full of borrowed greek words because Cicero and his friend were big greekboos, there are a lot of stupid inside jokes, puns, run-on sentences, etc. There's also the fact that this mail shows Cicero as someone very indecisive and sometimes awkward which is a big contrast to Cicero's public speeches which are always straight to the point.

>>2111330
Do we have any of Cicero's private mail between him and a lady? I recall that in Elizabeth of Bohemia's letters to Descartes both of them were very prim and proper, almost verging on flirtatiousness. I ask because I am wondering if people in the 16th Century wrote more formally than in Antiquity. I would also be interested to see some of Descartes's more informal private messages. Do you know if these exist? Or another person's from the era?

>>2111330
>to a level it's considered basic translation material

This is also important. Nowadays we mostly learn grammar through practical, simplified examples, but in various cultures in the past you learned to read, copy, even memorize important texts which tend to be formal in some way, such as a philosophical treatise, a collection of Classical letters, even religious texts.