DRAE doesn't say anything beyond that it comes from Latin. Wiktionary doesn't give even that. I even tried a bunch of other resources but nothing came out. Yes, I know it comes from Latin but I would love to see if anyone has anything more than that. I want to understand where the Latin platica came from and what it meant.
It seems that that latin form of plática was not a name originally, but the feminine form of and adjetive, which meant elemental or basic. It seems that it was used in expressions like sermo platicus, which means basic or rudimentary conversation, and disputatio platica, which means basic argument or discussion. Finally it turned into a name to define a chat.
Unfortunately I found no sources that confirm me, are just class notes.
The word plática derived from práctica in the Latin sense of "practicus", ("practical", "active"), which is related to the Greek "praxis".
In the usual sense, "práctica" refers to something habitual exercise. But "plática" refers to what is ordinary, usual.
In that sense, platicar originally refers to talk about what would be ordinary, common, current.