I'm not saying this is how it happened, but it's how I lived through it

When email was first invented, it adopted a simple way to figure out where to be sent. Physical addresses need several layers of information so that postal workers can figure out where to send something by focusing only in a single step of a hierarchy at any time (country, region, city, neighborhood, street, house). Electronic mail needs only 2 big pieces of information: a mail server (a single "computer") and the recipient in that server. Thus, an address was sent to, say, Andy at the UNAM server.

The convention settled in using the @ symbol to separate these two pieces of information, and so an e-mail address is generally read as "user at server".

Years and years later, Twttr came along. Since it was being used only by a few dozens of people, it made sense to keep a Public Timeline with all tweets being published in real time. Even though the information flow was millions of times slower than today, sometimes one needed to quickly signal someone else for attention, either to start or continue a conversation, to retweet (manually!) or just to acknowledge as someone being worthy of being followed.

The community settled in the @ symbol because it built on the existing notion of it being used to designate someone, similar to email. And so the idea of mentioning someone through an a symbol became so widespread that eventually was officially adopted by twitter (yes, @-mentions, hashtags for filtering results and retweets were all mostly features created by the userbase and only later made an official part of twitter)

The official implementation of @-mentions brought with it the necessity of also implementing a notification system: a way of seeing all mentions received so that one could engage in conversation, retweet or maybe even block.

And so the idea of @-mentions grew and expanded beyond Twitter, including services like Facebook, Instagram, Whatsapp and Discord. It became the de facto way to draw attention to a person.

Then, the memetic phrase "don't @ me" is often used after a declarative sentence (serious or not) to indicate that one is not willing to debate the issue, that one's opinion on the matter is final.

For example:

«Pineapple does belong in pizza. Don't @ me»
Of course, it's often used for comedic effect when one uses it after a very trivial matter; e.g. «vanilla ice cream is best ice cream don't @ me»

raincomplex says: there are parts in the bhagavad-gita where a guy will finish a grandiose rambling bit with "this is my opinion." and now i can't help but replace this in my head with "You are the supreme primal objective. You are the ultimate resting place of all this universe. You are inexhaustible, and You are the oldest. You are the maintainer of the eternal religion, the Personality of Godhead. Don't @ me."