The Vilna Shas is relatively recent (it was published in 1886). It was the most extensive Talmud ever printed at that time, including dozens of early commentaries printed for the first time from manuscripts, as well as later commentaires, all painstakingly edited by talmidei chachamim. It took them six years to prepare the new edition. As such it was a superior product, and was highly successful.
In parallel, their main competitor, the Slavita Shas had sold out by 1835, and for different reasons that printing house was shut down by the government.
Add to this the power of memorization (talmidei chachamim know where on the page specific citations/commentaries are) which helped make it the standard.
Interestingly, the printing of the Vilna Shas was done on a "Kickstarter-like model": due to the substantial costs involved in the editing and printing, the publishers issued a notice to the public announcing their Talmud, together with a sample, and asking for pre-commitments to purchase it when it appeared. Response was overwhelmingly positive and launched the venture. The high investment required was another barrier to entry for competitors.
Finally, soon enough, Europe got engulfed in World War 1 and following, which massively restricted printing of new books. New editions of the Talmud only restarted to be prepared after the end of WW2.
The factbase above leverages Akiva Aaronson's People of the book, a wonderful book about Jewish books. See also here for further reading.