TL;DR Mostly by circumstance and coincidence in the late 19th century/early 20th century. And since we've just continued the tradition.

During the American Civil War, President Abraham Lincoln signed into law what was called The Lieber Code (named after Franz Lieber, its author). This was the very first set of official, comprehensive, and codified into law set of rules and regulations for warfare/martial law. A copy can be found here

Within its pages it set out rules and regulations for a variety of things including: protection of civilians and civilian property and punishment of transgression; deserters, prisoners of war, hostages, and pillaging; spies; truces and prisoner exchange; the conditions of any armistice, and respect for human life; assassination and murder of soldiers or citizens in hostile territory; and a wide variety of other things.

In 1874, Tsar Alexander II called together delegates from 15 European States and they drafted what became The Brussels Declaration (found here). This borrowed heavily from The Lieber Code, but it was not adopted by all of the major nations so it had little affect on subsequent conflicts. In 1880, more meetings were held, this time in Oxford, England. These initiatives led to the publishing of the Manual of the Laws and Customs of War at Oxford in 1880 (found here. Again, not all of the major European nations adopted these customs.

Flash forward to 1899, the end of the 19th century, a century filled with various wars and conflicts as you can see, the first "Conference for Peace" was held in The Hague, Netherlands on the suggestion of Tsar Nicholas II and Dutch Lawyer Tobias Asser. The 1899 Convention borrowed heavily from The Lieber Code, as well as The Brussels Declaration and The Oxford Manual. There was a second Convention held in 1907 which focused primarily on maritime war and naval warfare (as this was increasing in the Pacific theater).

NOTE: There are much lengthier answers and information available that covers these specific Conventions but it's not terribly relevant to your question. For more information on what was passed and which countries adopted the Convention's treaties/declarations I suggest the International Humanitarian Law Database which has tons of resources and copies of documents. https://ihl-databases.icrc.org/en/

One major resolution from the two conventions was the establishment of a Permanent Court of Arbitration (PCA) which sought to settle the majority of international disputes without the need for war. Courts need courthouses. Andrew Carnegie was a very wealthy man and he donated the funds to build the Peace Palace which would house the PCA. This was built in The Hague as this is where the Conventions had taken place.

After WWI, which came about because of multiple violations of the Hague Convention documents, the League of Nations was formed. Again, without going too much into the League of Nations itself, the League established what was then called the Permanent Court of International Justice (now known as the United Nation's International Court of Justice). These Courts would also be housed in the Peace Palace as everyone agreed it was convenient as well as now customary at that point.

Further developments since the 1920s have taken place, like the establishment of The Hague Academy for International Law (1923), the opening of a United Nations office in The Hague (1946), the Iran–United States Claims Tribunal (1981), the International Criminal Tribunal for the former Yugoslavia (1993), founding of Europol (1998), and the International Criminal Court founding (2002). All of these have taken place in The Hague and have helped establish both the Palace and the city as the center for international legal arbitration.