Etymology of "thirsty" as slang for horny, covetous, desirous

It is actually a very old usage, thirst:

• Figurative sense of "vehement desire" is attested from c. 1200.

(Etymonline)

Green's Dictionary of Slang notes in their entry on this sense of "thirsty" the context US Campus, indicating that it is very recent slang used by U.S. college students and other young people. No actual date is attested.

2. (US campus) eager for sexual attention.

Other write-ups on millennial slang, like this one in Metro, sometimes refer to the word as being primarily about wanting attention, not necessarily of a sexual nature.

According to Corina Chocano of The New York Times, writing in April 2017, the term originated in black culture and then shifted to Internet culture. The term is described in depth in an article in The New York Times from April 2017.

“Thirst,” in recent black and then internet slang, describes a graceless need for approval, affection or attention, one so raw that it creeps people out.

Chocano draws attention to the growth in popularity of the term at the time of a 2014 Diet Coke ad with the tagline, “Be ambitious, not thirsty. You’re on Diet Coke.”

It's worth noting that thirsty in the 1980's to present has also been slang for desperately craving drugs, according to Green's Dictionary of Slang. This could have been a precursor for the meaning related to craving attention of a sexual kind or otherwise, and is probably part of why the Diet Coke commercial drew a lot of attention.

“Thirst” was first added to the Urban Dictionary in 2003, but its use hit its first great peak sometime around 2014, the year Diet Coke introduced the thirstiest ad campaign in history. One ad’s tagline seemed to suggest any number of strange things: that drinking soda was like snorting cocaine, that the average Diet Coke drinker was prone to using sex to advance her career, maybe even that she was plainly desperate and needed to be reined in. (“Be ambitious, not thirsty. You’re on Diet Coke.”)

There's no question of slang here.

Broadly to thirst for is to want.

To be horny is to want

To covet/be covetous of is to want

To desire/be desirous of is to want

"Thirty" is what you say about someone online when they post an over-sexualized picture of themselves. If a girl posts a selfie in a bikini, or a guy posts a selfie with his shirt off, for example, someone might reply with a meme calling that person thirsty. They're just looking for praise for their sexy picture and hence are "thirsty" for attention.

Someone who makes a post like that might get a response with a meme mocking them for their "thirst."

In Sopranos Season 1 episode 6, Carmela uses "thirst" like needing sex. That's back in 1999.