Why did it stop?
It hasn’t. The Bureau International des Expositions convenes world fairs (now more commonly “Expos”) every few years, along with “specialized expos” in particular topics. The last World Expo was held in Milan in 2015. The next one will be Dubai 2020.
They haven't stopped, they've just changed a lot. They don't focus on the "world of tomorrow" anymore and it's more of a cultural exchange/PR for your country type of thing. Recently the themes have been environmental in nature.
The most recent one was 2017 in Kazakhstan (theme was future energy) and the next is in Beijing next year.
CptBuck and Littleanomaly have already answered briefly. There have been other discussions before.
I think the best is "When did the worlds fair stop being a big thing and why?" at https://www.reddit.com/r/AskHistorians/comments/4nyuhm/when_did_the_worlds_fair_stop_being_a_big_thing/ . /u/DerProfessor started "Ah, my area of expertise!" and delivered.
See also "Why did world fairs stop being a big things?" at https://www.reddit.com/r/AskHistorians/comments/3nc9nb/why_did_world_fairs_stop_being_a_big_things/ , replies by /u/GravitasIsOverrated , /u/NeedsToShutUp , and MrDowntown.
"I have so many Questions about the World's Fair !!" at https://www.reddit.com/r/AskHistorians/comments/1jfq40/i_have_so_many_questions_about_the_worlds_fair/ : reply by mvlindsey and MrDowntown.
"What happened to the World's Fair? What caused it's decline in popularity and general loss of grandiosity?" at https://www.reddit.com/r/AskHistorians/comments/1ie5u4/what_happened_to_the_worlds_fair_what_caused_its/ : AskHistorians was not as strongly moderated then, so there is more "fluff", and some of the points are repeats from above. But I don't think TheEmperorsNewHose's points were covered in that way (and Diaiti wrote a bit about the rise).
To add to the answers already here, I wrote a bit about the World's Fair here. In the age of the telegraph and steam engines, World Fairs were ways for countries to flaunt their greatness to an assortment of visitors: Italy hosted one in Milan to prove that after unification it had come into its own as an industrial power (and this is the fair I'm most familiar with; given my interest in Italian history and economics).
Nowadays news, people, and even goods travel very fast; we don't really need to travel to a World Fair to see the latest ideas and technologies developed by inventors in faraway countries. However, this hasn't stopped fairs from being staged: recently, Universal Expositions have been used to redevelop cities as well and mark the start of large public and private construction projects. In 1992, for example, the Spanish government staged a Universal Exposition in Seville to mark 500 years since Columbus' voyage. The Exposition yielded a flurry of public and private construction and redevelopment.