I left a quick comment on your other post on /r/linguistics but I've got a few more minutes now and can expand on it, and happened to see this here. Gonna borrow a bit from an older answer to a similar question.

First is that the terms "Ancient Chinese" and "Archaic Chinese" are no longer used. Karlgren coined and used these, but we've moved on, and now talk about Middle Chinese and Old Chinese. Middle Chinese is what was spoken around the year 600CE. There have been a great many changes to the language varieties since then, much like Latin compared to French or Spanish etc. That is to say, an average monolingual Mandarin going back in time to Chang'an in 600 would probably be able to tell the language spoken there was related, but they wouldn't understand what was being said on their first day.

Around the 1200s, Mandarin was pretty well identifiable as Mandarin. As in, we'd recognise it today as such. We refer to the language of the 1200s spoken in the north as Mandarin. It was still somewhat different than many of the Mandarin dialects spoken today, most significantly Standard Mandarin.

Before this, we could call it Proto-Mandarin, and we do talk about Early Mandarin, as the thing spoken around the time of the Yuan. Before this it's a bit harder to say. The Song Court spoke Mandarin and brought it with them as they fled the Mongos with the fall of the Northern court and resettled in Lin'an (Hanzhou). This same period is what brought Mandarin to Nanjing and Hangzhou more generally, displacing Early Wu.

The "how" part of your question is a little more boring, because the answer is basically just "language changes". Tones merged, moving from the old 8 tone system (though really 7 since the widespread shang/qu merger happened quite early) down to a 3-5 tone system (depending on where you were), losing the ru tone category completely with a few dialectal exceptions. Palatalisation (Nanking -> Nanjing) was widespread. Diphthongs were retained where lost in other varieties. And then of course there was a lot of mixing around of things thanks to various efforts to standardise, going from the Late Qing through to the Nationalist control in the 1930s.

That's the simple explanation. I hope that helps answer your question a little better. Let me know if you have any follow ups or want clarification.