Because the ideas were spread throughout Europe and then the world and shaped the 19th century and beyond. The revolutions of 1848 were a reflection of the French Revolution. The Unification of Italy, Romania, and Germany were reflections of this. World War 1, I argue, was the most violent reflection of the French Revolution.
One of the big issues of the Revolution was the idea of Nationalism, that is people were members of a nation, not just subjects to a monarch or a city prince. They idea of being a Frenchmen, a German, an Italian, etc emerged violently. This idea formed the German Empire, yet wiped out the Austro-Hungry empire and the Russian empire. It was also one of the major factors in the US Civil war.
The Social ideals of the French Revolution, "liberty, fraternity, equality", paved the way for acceptability of many ideas that the Enlightenment thinkers had been working on, and coupled that with the energy and gusto of the romantic period.
I hope this helps.
Because it was the first practical application of Enlightenment political philosophy in an Old World monarchy (rather than in a colony like the USA), and because it marks the beginning of the end for absolute monarchy and the divine right of kings. The execution of Louis XVI is an event that still resonates in our political consciousness - it marks the replacement of God by Reason and the common will of the people as the main source of political legitimacy (as described by Rousseau in The Social Contract). Fifty years after 1789, there was no absolute monarch left anywhere in Europe except from Spain.
This period is also fascinating because it's full of political innovations and upheavals - France had twelve different Constitutions between 1789 and 1852, including the Revolution, Republic, Empire, another Republic, another Empire, and that even discounts non-constitutional events such as the Commune (probably my favorite historical subject and one of the only historical examples of a "successful" anarcho-federalist political organisation), after more than a millennium of monarchy. Taking the longer view, the transfer of power that took place between the aristocracy and the bourgeoisie has been analyzed (by Marx among others) as one of the factors presiding to the birth of modern capitalism, setting the stage for the Industrial Revolution.
The Revolution and subsequent events (the Terror, the Commune, the Empire, etc) are crucial factors in understanding not only European and world history, but also the history of ideas. Hegel and Marx were heavily influenced by their study of the period, for instance, and that in turn had an extraordinary impact on the first 20th century.
Source: Albert Camus's L'homme révolté for the philosophical side, and Francis Demier's La France du XIXème siècle for the history.
It was such an important events for many reasons. The first, is that France was one of the world's largest superpowers and had a massive economy. Imagine if China had a sudden rebellion, with members of the National People's Congress being beheaded on the steps of Rénmín Dàhuìtáng. Major companies falling apart overnight, and a fragmented military combating massive armies of angry citizens. The world's economy would be quite shaken, and a lot of people would die. Though globalized economics wasn't what it is today, this is the closest you can get to a modern understanding of the French revolution. France's economy had grown quite quickly in the 1750's, due to expanding need of mines, metallurgy and textiles. In addition, France developed a flourishing agricultural industry. Though the Seven-Years War did put a damper on the French economy, it had still had seen tremendous growth since the beginning of the 18th century. Mechanization was being introduced, monopolies owned by the ruling class were growing in size, and factories were popping up across France. And though poverty in France was quite high, France's global economic output was also quite high. Indeed, France was the largest exporter of slaves in the world. France was doing business with England, Spain, Portugal, the Italian states, India, America, and Antilles. Yet this would begin to fall apart in last quarter of the century, with poverty reaching all time highs and many French industries beginning to bottom out. And as with any falling economic giant, there were shockwaves. And these waves were felt in every single continent as French industry collapsed under its own weight leading up to and during the French revolution.
This is where the second largest factor comes into play. With France being such an economic hub in the 18th century, they had a large area of influence. With the American ideas of liberty, freedom, and equality, it gave hope to the French that they could rise against their masters out of the depths of serfdom. And when these ideas spread through France, they spread through the entire region. Like a plague, the ideas of freedom and liberty would strike the oppressive governments of Europe even into the 19th century and opened society to the philosophies of the enlightenment. Philosophies proposed by men such as Jean-Jacques Rousseau , François-Noël Babeuf (who, along with Rousseau would have great impact on sociology and would inspire future philosophers such as Karl Marx), and Jean-Baptiste du Val-de-Grâce saw a new freedom to spread like wildfire.
So in essence, the French Revolution saw an upheaval to one of the (at the time) most established economic superpowers, as well as new host philosophies which would have major impact on the world.
Sources:
There are many good answers already, but I want to add to this that the revolution transformed France administrative structure into something resembling a modern state. Almost a thousand years of bloated, evergrowing feudal, religious and bureocratic traditions was swept away by the revolutionary government and replaced with secular and centralised state power. This in turn forced the other states of Europe to adopt similar reforms to be able to compete, and voilà, a violently modernized Europe in a matter of decades.
tl:dr It forced the birth of the modern nation state.
I would recommend that you look here as to the importance of the French Revolution (at least compared to the American Revolution).
In addition to the wonderful answers, it also gave birth to the universal system of measurements. It is hard to overstate the importance of such a feat.
http://physics.nist.gov/cuu/Units/history.html
I found this episode to be absolutely fascinating. http://www.radiolab.org/story/kg/
I'd say that the french revolution of 89 and the second of 92 (even more important) are part of something that starts from the english revolution, continues to the american one and this whole thing goes to 1830/1848 and is at end in France in 1870's after the royalists are crushed definitely.
Every revolution since 1800 was made under the sound of the french anthem "la marseillaise" (even in 1917) and was inspired by thinkers like Sieyes or Condorcet.