Great question!
First off, it's important to note that The Dutch Republic wasn't a true republic in the modern, post-American Revolution, sense of the word. It's best qualified as an oligarchical republic, in which the oligarchs were a relatively numerous group of wealthy burghers / bourgeoisie and gentry which, through local and state government effectively controlled the Dutch Republic in (among themselves) a more or less democratic fashion. Throughout the Republics existence, there were two basic political factions: the staatsgezinden (Republicans / literally "state favor-ers") and the oranjegezinden, those who supported the maintaining or increase of the powers of the descendants of William of Orange (the initial leader of the Dutch Revolt) who occupied a function (the de facto head of state) referred to as the stadholder.
Technically, this was a position which had evolved from the medieval practice of appointing stewards; it literally meant 'place holder'. In the case of the Dutch Republic, every province appointed its own stadholder though usually, several provinces chose the same stadholder and there rarely were more than two at any given moment. Most importantly, the stadholder of Holland and Zealand commanded the army and navy of the entire Dutch Republic.
At first, it wasn't an hereditary title but because of the prestige and family wealth of the first stadholder (William of Orange, the leader of the Dutch Revolt) the stadholder office was almost ubiquitously held by a member of the House of Orange Nassau. During the latter part of the 18th century, the then stadholder William IV made the title hereditary, which had already effectively been since its inception. His son, William V, was the last stadholder of the Dutch Republic.
William V reigned in the aftermath of the American Revolution, in which the Dutch Republic had supported the American Rebel against their common enemy the British. The unintended consequence of this (as in France, which had done the same) was a rise in 'true' republicanism in the Netherlands and support for William V crumbled as so-called patriots (a new, more extreme wing/evolution of the staatsgezinden) were taking over power in the Netherlands via a kind of slow, low intensity civil war that began in 1781. He was only to remain in power by appealing to this father in law in 1787, Frederick II of Prussia, who was willing to send additional soldiers, forcing the patriot faction to exile itself in France, mostly in the Northwesternmost part of France, near Calais/Dunkirk; which was mostly Dutch-speaking at the time. After/During the French Revolution, William V was part of the First Coalition against France in 1793, but in 1795 the Netherlands were invaded by the Dutch patriot exiles and the French Revolutionary Army; leading to Williams exile in Great Britain (where he died in 1808) and the establishment of the Batavian Republic, a sister republic of France.
His son, William VI, claimed the hereditary stadholdership but when the British signed the treaty of Amiens in 1802 (recognizing the Batavian Republic thereby effectively undermining his claim) he aligned himself with the Prussian, Austrian and Russian courts; fighting against Napoleon on various occasions; where he generally made a good impression on his superiors and was especially well liked by Alexander I, the Russian Czar.
Meanwhile, in 1806, the French effectively annexed the Batavian Republic and created the puppet Kingdom of Holland, ruled (as King of Holland) by Louis Bonaparte, one of Napoleons brothers. After being judged a failure by his older brother, the Kingdom was formally annexed by the French Empire (the Netherlands being, in the words of Napoleon, "no more than the alluvial deposit of French rivers") in 1810.
As Napoleons Empire collapsed following the battle of Leipzig in 1813, William VI returned to the Netherlands where (in the aftermath of 7 years of French annexation) he was offered the title of King. Seeing this a French invention, he refused and instead said he would "only" take the title of "Sovereign prince of the Netherlands"; which made him king in all but name. Due to the disastrous invasion of Russia and subsequent Russian offensive, czar Alexander was one of the key players at the Congress of Vienna where he lobbied for the creation of a strong buffer state on France's northern border: the United Kingdom of the Netherlands, of which William IV (as William I) became king, betrothing his son to a Russian princess in the same year.
The King himself was warmly welcomed by most of the population, as the French had thoroughly outstayed their welcome and had ruined the Dutch economy. His absolutist rule less so, especially by his French-speaking ones in what is now Belgium; who revolted in 1830 mainly to this style of governance, his Protestantism and predisposition towards his Dutch-speaking subjects; forming neighboring Belgium in the process. During the Revolutions of 1848, his son, William II, turned the Netherlands into a constitutional monarchy rather than risk another revolution; thereby securing the continued existence of the Dutch monarchy until this day.
So in short;
The transition from Republic to Monarchy was not as black and white as it might appear. The first Dutch king William I, came from a long line of stadholders who had wielded powers quite similar to those of a hereditary monarch during the time of the Dutch Republic for centuries. Following 1815 the title changed and the powers coming with it increased, though not to a truly massive extent. Nevertheless, during the counter revolutions of 1848, liberal forces reshaped the country into a constitutional monarchy.