The French Revolution affected other countries because there was immediately a perceived threat that the French would attempt to spread the Revolution to other countries. The movement to overthrow, or at least to limit, the power of the monarchy was, at the time, thought of in similar terms to the perception of communism 150 years later: namely, that this was a plan to take over the world and to convert countries to the same anti-monarchical form of government the Jacobins in France were attempting to set up.

Virtually all of the states of Europe in 1789 were monarchies, and the only large country that had a degree of genuine democracy as a constitutional monarchy was Great Britain. Given that egalitarian principles had been expressed again and again throughout the eighteenth century by Enlightenment philosophers and political theorists, and given the example of democratic government which had begun just a few years earlier in America, it was understandable that European kings and queens would become more than a bit paranoid over what was occurring in France. In addition, there was the personal connections of Marie Antoinette, as she was the sister of the Holy Roman Emperor Joseph II (and Leopold II after Joseph's death in 1790).

Yet it was not until 1792 that hostilities broke out, with Austria and Prussia first declaring war on Revolutionary France. As long as no actual harm had come to Louis XVI and Marie Antoinette, the other powers had taken no overt action. The first decisive change occurred in June 1791, when the king and queen attempted to escape across the frontier into Germany and were halted and then placed under what was essentially house arrest. It was clear that the king was now a prisoner in his own country. From this point, the series of events including the king's being forced to accept the constitution in September of 1791, the abolition of the monarchy a year later, and the execution of both the king and queen in 1793 meant that a state of irreconcilable conflict existed between France and the other European states. What followed, of course, was over 20 years of nearly continuous war until the final defeat of Napoleon in 1815 at Waterloo.

The French Revolution had a massive impact on the world.  One of the more well known impacts is the domino effect of revolutions inspired by the French Revolution.

In Haiti, which at the time of the French Revolution was a French colony called Saint Domingue, independence movements broke out when the National Assembly passed the Declaration of the Rights of Man and the Citizen.  If all men are free, could that include the enslaved Africans in the French colony of Saint Domingue?  Initially, all men in Haiti were granted their freedom, until Napoleon's need to finance his wars in the early 19th century saw the reinstatement of slavery of Haiti.  After a few years of fighting, the Haitians won their independence from France, a movement directly caused by the French Revolution.  When Haiti became independent, revolutionary leaders in South America like Simon Bolivar were inspired to fight for independence in many South American colonies, and most achieved their independence by 1825.

In Europe, other nations that had similar social problems and political structures of revolutionary France staged similar liberal independence movements.  Some countries who attempted revolution include Italy, Prussia, Austria, Germany, and France (again).  These are called the revolutions of 1848 and were inspired by the French Revolution.  The idea was that if the citizens of France could overthrow (and execute) their King, why can't the citizens of other European nations?  While monarchs fought hard to suppress this thought, and in most cases won, it still shows how great an impact the French Revolution was on Europe.

After the French Revolution, Napoleon Bonaparte staged a coup d'etat and took control of France.  During his reign as emperor, he expanded the French Empire across Europe, challenged British overseas territories, and ultimately bankrupted France.  While his rule was not successful, and he was ultimately replaced by a King (negating all of the ideas laid out by the third estate in the French Revolution), he was a product of the French Revolution.  The French Revolution's principles were spread throughout Europe because of Napoleon.

The French Revolution impacted other countries in various ways.  In Europe, the revolution led to a series of wars between various countries and the French.  It also led to the rise of Napoleon and the empire he won, albeit briefly.

Since you originally put this in the "United States of America" section, though, I will look mostly at the revolution's impact on the US.  The French Revolution caused the US a great deal of trouble.  First, it helped split the country between pro-British and pro-French factions.  Second, it led to many problems in foreign policy.  The French and British went to war, and the US was caught in between with both countries preying on US shipping to the other country.  This eventually led to near-war with the French and finally to the War of 1812 with the British.