There is a four major points.

First, it was adopted by imperial germany Kriegsmarine and Heer a few years prior Great War. Germany per se was one of biggest firearms producer in world and when Great War ends, Germany left lot's of pistols and ammo stockpiles which they sold nearly everywhere at low prices, so other countries buy german surplus for their own militaries for many reasons, USSR use 9 mm Luger with Mauser C96 "Red Nine" pistol mostly because Germany was just a few countries which want trade with them, Poland introduce it due to cavalry demand for sidearm with strong enough cartrige to kill wounded horses, France announced tactical and technical requirements for a new pistol, resulting in the creation of the Browning High Power pistol using 9 mm Luger in 1935.

Germany also introduce "gamechanger" on battlefield called machine pistol in late stage of Great War. 9X19 mm offer enough reliability to construct quite simply blow-back design (basically you put strong spring on one end of gun and bolt on another end, cartrige pressure push bolt back and spring bring back it on place every time) machine pistols.

Another point came after WWII, newly created NATO start standardize equipment between NATO members. In 1982 NATO create STANAG 4090, a standarization document for 9X19 mm, to ensure functional interchangeability of this ammunition on the battlefield. De facto it made 9 mm Luger NATO standard caliber for all sidearms and machine pistols. Other American allies reliant on NATO members production cababilities and american military aid follow suit and finally there was no other caliber on market because anyone use it already.

Finally, in '80s "Wonder Nine" pistols appear on market. Whole list of pistols using 9 mm Luger with high capability magazines (10 rounds and more), quite compact and lightweight and affordable. Basically all you need for a sidearm for police, military or security.

Today, while there was a attemp to replace 9 mm Luger after some police incidents like FBI Miami shootout, where eight FBI agents had problems to stop two armed bank robbers due to market saturation with guns and ammo producers it still is one of market standard.