The Dutch didn't monopolize global trade. The biggest part of the wealth was coming from the Baltic Sea trade. Which was lucerative but reasonably close to the Dutch Republic. As such it doesn't really speak to the imagination as much as the voyages to the East Indies do.

Now to answer your question of how the Dutch got into the position to dominate the trade from the Baltics and Indonesia. It was a set of lucky circumstances and clever financial systems that got them there.

During the Eighty Years war the united low countries saw themselves in conflict with both present day Spain and Portugal. Portugal at the time had a monopoly on the trade in the East Indies. This was a very profitable endevour and competing for that trade would both increase the wealth of the Low countries as well as hurt the enemies ability to wage war due to lost revenue. The problem was nobody besides the Portuguese really knew how to get there.

This changed when Jan Huygen van Lindschoten published his book Itinerario which thorougly explained the voyage. In Dutch secondary schools we teach he "stole the maps" to the West indies. Maps at the time were more like books explaining the voyage in great detail. "Embark there and retrieve fresh water. Watch out for the coral reefs there and there" Make sure to hunt seals when you arrive at .... etc" Now it was finally possible to contest the West Indies and their riches. The Spice Wars began which is really just a part of the Eighty Years War in my mind.

During the Spice Wars the Dutch were largely succesful in driving back the Portuguese from the East Indies. Please note that this isn't the era of total colonial domination yet. The wars were fought for a few fortified positions in Dutch: factorijen. From which the spices were collected and guarded for the next arrival of ships to take them back to Europe.

A big part of the succes of the Dutch Republic lies somewhere else however. The Republic enjoyed a population with a large amount of capital waiting to be spend. Partly because of the blockade of the Schelde river. Bruges (edit: This should've been antwerp. Bruge was already on the decline at that point). was one of the most important trade cities in medieval times. Actually Flanders as a whole was a very rich region in medieval times. This region was however, entirely dependent on the Schelde river to export their textile goods. When the river was blockaded by Dutch forces many rich, experienced craftsman and merchants were incentivized to set up shop in cities in the Dutch Republic. This proved to be an important economic boon for the Republic. Another part of the exodus of present day Belgians to the Dutch Republic was because of the relative religious freedom in the Dutch Republic.

This however didn't just hold true for Belgians. The Holy Roman Empire was being torn apart by the Thirty Years War. French Hugenots were hunted down in France when the Edict of Nantes was repealed and England was tormented by religious wars. Many religious minorities fled to the safety of the Dutch Republic providing additional manpower and wealth. Often these refugees were rich, skilled people.

Now these were all factors for the succes of the Republic. Another big part, perhaps event he biggest part, of their succes in trade lies in this: naval wars aren't won through superior manpower. They are won through superior finances. And we just learned why the Dutch Republic was so rich in the first place.

Farmers in the Dutch Republic had the luxury of mainly growing cash crops. The Baltic Sea trade provided the Dutch republic with more grain than they had mouths to feed. This trade in grain and other bulk goods from Eastern Europe through the Baltic Sea was actually so important it was called the moedernegotie. In English it would translate in to something to the lines of Mother of all trades.

Succes in the Baltic Sea trade relied on the Fluitschip. You see each ship navigating to and out of the Baltic sea must pass the Kattegat. It was there that the Danish king levied tolls on each passing ship. The price of the toll was calculated on the size of the deck of the ship. Fluitschepen (fluteships) had a small deck but very large storage capacity beneath. This enabled Dutch merchants to haul vast amounts of resources for relatively low tolls. The production process of these ships was also very efficient and almost industrial through the use of mechanized labour in the form of windmills.

As said before naval battles rely mainly on money. Outfitting ships is very costly. In order to spread the risks of a failed trade expedition the Dutch Republic set up Compagniën (this is were the English word company comes from). You could pitch in a small amount of money to outfit a ship. The ship would make its voyage and, if succesful, you would receive your part of the profits based on the amount of money you pitched in. From there it is only a small step to a stock exchange as you are basically buying stocks from a ship and receiving dividend when the ship is back in port.

Now due to all these factors the Dutch provinces became their own biggest rivals in the overseas trade. As rich cities all had their own Compagnie. They were hauling so much spices that prices began to plummet. In order to keep prices artificially (edit2: low high) the Dutch Republic created the VoC. Vereenigde Oostindische Compagnie (Literally translated: United East indies Company). Which had a monopoly on all voyages to the East Indies. Less voyages means less spices means higher prices and more profits. The VoC was practically a state within a state. They were allowed to declare their own wars in East Asia. Set up their own governers, laws, make trade deals with local rulers etc. Quickly the VoC became very powerful.

The West Indian Company (WiC) was founded with the same aims for the West Indies. They didn't have nearly as much succes as their eastern counterpart. The aims of the WiC were mainly to disrupt the flow of goods from the Spanish colonies to the Spanish mainland. They also were a player in the slavetrade, but a rather small player compared to the Spanish and Portuguese. As such I will not go into great detail.

The decline started when and this is going to sound rather crude, other nations got their shit together. The French and mainly the English were, with their much larger populations, always predisposed to take the crown of the Dutch Republic. It was a series of fortunate events that led to their Golden Age. But once the religious turmoil died down in Europe and French and English got a sense of the money that was being made by the Dutch trade practises, it was a matter of time before they copied what made the Dutch succesful and won through economies of scale.

This has become quite a rambling and while I am absolutely sure that all of this is factually correct I might have made some errors in the chronology. So I hope this is an acceptible answer and if someone more knowledgable than me can expand that'd be great.