“Koningsdag” translates to “King’s Day,” and marks the birthday of the current King of the Netherlands, Willem-Alexander. The celebration has its roots in the 1880s, when the government began celebrating the monarch’s birthday as a way of promoting national unity. (Amusingly, the first holiday was actually the Prinsessedag on August 31st, 1885; no one liked William III enough to celebrate his birthday.)
Many attributes of the arms of the Netherlands are directly derived from those of the House of Nassau, since William I, the first king of the Kingdom of the Netherlands, was of that house. The lion rampant, the field billette, and the tinctures are all directly derived from Nassau. The crown is an interesting case. The House of Nassau split into two branches in 1255 between the elder brother, Walram II and Otto I. At the time, Walram added a crown to the lion in his arms as a form of differencing. However, William I was not descended from the senior Walram line, but the junior Ottonian. The crown in the Dutch arms, like the sword and arrows, is borrowed from the former arms of the Dutch Republic.
The arms of the Dutch Republic (or a lion crowned gules armed and langued, holding in the dexter paw a sword and in the sinister seven arrows azure) came into use in 1584. The tinctures were swapped around near 1665 to be gules a lion crowned or armed and langued, holding in the dexter paw a sword and in the sinister seven arrows argent. The tinctures were from Holland, the most important of the provinces. The sword represented defense of liberty, the crown their newly-won sovereignty, and there was supposed to be one arrow per included province. This changed quite a bit, depending on how many provinces were members at any given time; although it eventually settled on 7 in 1606, the seal wasn’t officially updated until 1795. A shame they took so long, because the other thing that happened in 1795 was the Batavian Revolution, which established the Batavian Republic.
The only reason I can honestly say that the Batavian Revolution didn’t completely butcher the Dutch arms is because… the Batavian Republic didn’t actually use arms. They had an allegorical image of a “maiden of freedom,” which was apparently a resurrection of some of the nationalist symbology of the 16th-century Dutch Revolt. It did involve a lion, since that was now a well-established regional symbol, and a lot of Roman imagery, and a pole with a “cap of liberty.” It was mercifully short-lived. In a rare case of the Napoleons making sensible and visually attractive heraldic decisions, when the Kingdom of Holland formed in 1806, they quartered the Dutch Republican lion with the Napoleonic eagle, but that only lasted through 1810, when the kingdom was abolished. In 1813, the French (finally) got kicked out, and William – then the sixth Prince of Orange of that name – was proclaimed William I, Sovereign Prince of the Netherlands. He quartered the Dutch Republican lion with his personal arms, with Nassau (sans crown) in an escutcheon of pretense.
The Dutch arms took on their present form in 1815. Since the new United Kingdom of the Netherlands included territory that hadn’t been a part of the Dutch Republic, the former red-and-gold lion was no longer appropriate. Instead, William took the crown, sword, and arrows and added them to the lion of Nassau, resulting in the same arms the country uses today.