Explain how war broke out between the empires of Blefuscu and Lilliput?

The war between Lilliput and Blefuscu erupted over the issue of the proper way of breaking an egg. When the Emperor of Lilliput decreed that eggs should (contrary to the old practice) be broken at the small end, those people who clung to the old practice of breaking at the larger end were exiled, and were then given sanctuary and supported by the Blefuscuans.
It's not hard to see that Swift is parodying the kinds of religious disputes that constantly arose among Europeans (and others). Swift was a devout Anglican and a clergyman himself. One can interpret the Lilliput-Blefuscu dispute in various ways, especially as a parable of the disagreements between Protestants and Roman Catholics, or the disputes within Protestantism. But in spite of being a religious believer himself, one might argue that Swift seems to be attacking religion overall, not just a particular sect or denomination.
Swift's satiric approach is often to make a sweeping assault on human nature in general, rather than just specific factions or beliefs. In the Lilliput-Blefuscu war it is the universal silliness of humans who argue about trivial matters—not simply concerning religion, but anything—that Swift parodies.

Blefuscu and Lilliput are two neighboring countries of tiny men in Gulliver’s Travels. The war between these two empires starts due to a decree in Lilliput that bans an ancient practice. Traditionally people break their boiled eggs at the larger end, but ever since one day the king's grandfather accidentally cut his finger while breaking egg this way, this method of breaking eggs has been banned by the lawmaker in Lilliput and everyone must break eggs at the smaller end. The new law results in large-scale civil rebellions and lots of bloodshed. Many citizens of Lilliput who are against this law flee from Lilliput and seek refuge in Blefuscu, where the emperor allows them to break eggs whichever way they like. Blefuscu accuses Lilliput of not following the religious doctrine that "all true believers break their eggs at the convenient end" and attempts to solve this dispute through diplomatic negotiation but fails. Eventually, the war breaks out between these two countries. It has lasted three years so far.