It is going to be very difficult to provide a well-sourced answer to this question (partly because I don't speak Hungarian and partly because there is no direct answer to this question), but I will do what I can and provide additional sources if requested.

The first part of the reason for Hungary's dominance in water polo has to do with a strong, historic spa and aquatic culture in Hungary. Thermal water can be readily found in 80% of Hungary's territory. This ready access to thermal water has created a spa culture in Hungary that dates back 2,000 years or more. Similarly, the landlocked country (not always, see comment below) has access to two rivers and a major lake that provide ample opportunity to practice water sports. This culutre provides an historic and cultural basis for Hungarians affinity for water sports (widespread access/exposure to water sources).

Water polo began in England and Scotland in the 1870's. Water polo's rules were first codified in 1876 by William Wilson. Water polo would have first been brought to the attention of Hungary (as part of the Austro-Hungarian Empire) due to an exhibition game played by a team from England and a team from Vienna in 1896, but the game was formally introduced by Dr. Arpad Fuzzesery in 1899 when he brought a ball and rule book to Hungary. Within four years, the first Hungarian national championship was contested.

Club water polo grew in Hungary over the ensuing years and Hungary first competed in the Olympics in water polo in 1912 (but didn't medal). The reputation of Hungarian water polo as a powerhouse has its beginnings in a tour of England by the Hungarian club FTC in 1914; however, Hungary did not compete in the Olympics in 1920 and failed to medal in 1924. WWI had an impact on Hungarian water polo in that many players were lost to the war. Finally, in 1928, Hungary won the silver medal. Since this win, Hungary's national team has been one of the most dominant forces in international water polo.

It was also around this time that Hungary begin to influence the rules and tactics of water polo as well. A Hungarian is credited with introducing the "dry pass" to water polo in 1928. Furthermore, Hungary is credited with popularizing the over-arm stroke in water polo as well as tactics that encouraged mobility from all players in the pool. These innovations (and the manner in which they placed Hungary at the forefront of tactical development in water polo) greatly contributed to Hungary's historical dominance of the sport.

Lastly, it bears mentioning the fabled "blood in the water match" at the 1956 Melbourne Olympics and its relation to the Hungarian Revolution of 1956. While water polo could be considered the de facto national sport of Hungary by this time, this game further entrenched Hungarian water polo as a point of national pride and identity for Hungary. I would suggest the wonderful documentary about this game, Freedom's Fury, to get a feel for its larger impact.

So, while there is no one direct answer for why Hungary has been historically dominate in water polo there are a number of factors that have contributed to its place as the national sport of Hungary. A culture of aquatics and water sports, early adoption and innovation of the sport and the position of the sport in relation to the history of Hungary have all contributed to Hungary's profile in international water polo.

Lastly, it should be noted that the Baltic nations of Serbia, Croatia and Montenegro have overtaken Hungary as the dominant force in the sport in recent years. Rule changes that favor different skill sets and the declining profile of water polo in Hungary are usually cited as contributing factors to this decline.

Source: A Short History of Water Polo