No Girls on the Internet
The phrase originates from the godawful Early Days of the Internet, where it was seen as the dominion of basement-dwelling, generally male nerds. Most of these have since migrated to /tg/, so the meme fits perfectly.
Memes, being living breathing documents, are constantly evolving. Consequently, 'No Girls on the Internet' has gone through several different stages during its multidecade evolution.
Dawn of the internet
This view is a literal interpretation and is considered the origin of the meme. Men are generally early adopters of any technology or social structure. As a result, the early days of the internet contained an extremely small number of women. The phrase was therefore a objective observation of the state of the internet population.
As online games became more popular in the 90s, people began to discover that a good way to get free shit was to pretend to be female. Various idiots would give clan status, items, game currency, and other valuables to anything with a uterus in the hope that they would receive female attention. The meme evolved into a phrase to protect guildmates and friends from giving away their hard earned items to random fraudsters or from mistakenly engaging in ERP with other dudes.
Tits or GTFO
As the internet became more mainstream during the early and mid 00s, it was revealed that women DID use the internet. Unsure of whether a self proclaimed female was legitimate or not, people started to demand proof in the form of photos. These women discovered that they could get alot of attention simply for being female. This phenomenon manifested on 4chan as the boards were spammed with thread after thread of camwhores or trolls looking for attention. The spam began to displace high quality threads much like how carbon-monoxide displaces oxygen. Suffocating oldfags attempted to discourage the attention whore cancer by abrasively demanding that any girls display their tits or GTFO. This attempt failed miserably as the camwhores complied with the request and thus made the spam threads more popular than ever before.
Newfags began to argue with oldfags over the hostility and abrasiveness towards the camwhores. Soon women began to exploit their possession of uteruses (uteri?) to give their posts and arguments greater weight in the eyes of legions of newfags. As a forum that despises identities, 4chan decided to fight back against the cancer of female shit posters who started every argument with "Im a girl and i think that...."
The result is one of the best posts ever made to 4chan: