The short answer is that the Dutch Republic joined the First Coalition of Austria, Prussia, Great Britain and Spain in their war against Revolutionary France. The long answer is hopefully a little more interesting.

The first thing to realize is that the Dutch Republic in its political institutions was nothing like a modern Republic. Throughout the history of the Republic there had been a conflict between supporters of the 'regent' class of public officials and rich merchants on the one side, and supporters of the House of Orange on the other (generally lower middle-class and more religious people). The House of Orange always occupied the position of 'stadhouder', the highest office in the land which most importantly came with command of the armies and the navy. There was always a struggle between the Stadhouder and the provinces, particularly the province of Holland, making up the Dutch Republic about how much power the stadhouder should have exactly and how much money, but also between the provinces about whether the navy or the army was more important, and so on.

In the 18th century though, the battle lines had shifted. While previously the Stadhoudership had been an appointed position that only in fact was always awarded to the Oranges (leading on two occasions to no stadhouder being appointed at all when no suitable Orange candidate was available and when the previous one had misbehaved too much), in the 18th century the Stadhoudership formally became hereditary for the Orange family. The stadhoudership also accumulated rights to appoint representatives in the many, many councils through which the states and cities of the Dutch republic were ruled. Members in these councils, for cities, guilds, regions, water boards etc. were usually not elected either, but coopted new members, which is to say that members of the ruling class selected their own relatives and friends to help perpetuate their own hold of power .

So the Dutch Republic was not very democratic, but perhaps more importantly, its rulers were also not very competent. The army and navy were neglected, the last Stadhouder, the prince of Orange William V, was corrupt and incapable, and influenced far too much by the German duke of Brunswick who commanded the armies for him. The economy had stagnated and industry was in decline, though the Republic was still an important financial center. So, because of this and as a result of Enlightenment thinking, and also inspired by the American Revolution, a group of people started to protest the corrupt and oligarchic elites. These people, were known as the "patriots" as opposed to the "prinsgezinden", the supporters of the Prince of Orange.

The patriots eventually revolted and chased the stadhouder out of Holland in 1787. The stadhouder however was marrried to the daughter of the king of Prussia, and when she was detained trying to reenter Holland to negotiate with the Estates General there, he sent in his army and chased out the patriots. Many of the patriots fled to Paris, where many of them enthusiastically supported the French revolution and the idea that the revolutionary French could liberate the Dutch Republic from the clutches of the Orange family.

So when the French revolutionary army under general Pichegru defeated the small and scattered Dutch armies and crossed the frozen rivers to conquer all of the Republic in 1795, they could claim with some justification that they were 'liberating' the people of the Netherlands. The French army was accompanied by armed brigades of "patriots" and may have had help in overcoming the defenses from patriots who remained in the Republic and escaped the Orangist repression. After the surrender of the Dutch Republic, these patriots set up the "Batavian Republic", with the purpose of setting up a far more democratic system. However, for the French this Batavian Republic mostly had to serve as a client state to provide money and troops for their further wars. Eventually it was abolished by Napoleon in 1806.

Primary source for this comment is Simon Schama's Patriots and Liberators: Revolution in the Netherlands 1780–1813 (1977).

Further source is this 1893 review in De Gids of a dissertation on the 1794-5 war between the Republic and France.