How were human rights influenced by the French Revolution?
For an example of how the French Revolution impacted human rights in their colonies, I direct you to the Haitian Revolution.
Haiti was an island colony that was very profitable once the French convinced Caribbean pirates to settle down there (yes- there were actually some pirates of the Caribbean...just not the Johnny Depp kind). At this time, it was known as "Saint Domingue". Saint Domingue produced coffee and sugar, making Europeans happy with the new products and making France very rich. By the 1760s, Haiti was without a doubt the most profitable colony in the Americas. One was to keep profits high was to import slaves from Africa- not having to pay wages meant more money for the plantation owners.
When the French Revolution broke out, the slaves broke out in rebellion in St. Domingue. A long story short- the French decided to grant freedom to the slaves in their profitable Caribbean colony. The first part of the Haitian Revolution was successful! As it states in the Declaration of the Rights of Man and the Citizen, "Men are born and remain free and equal in rights." This also applied to the African now-made citizens in St. Domingue.
Later, Napoleon reinstated slavery to make more money to finance his wars, which brought about the second part of the Haitian Revolution. As a result, St. Domingue was let go and made a free country, the second in the Americas after the United States, in 1804.
So, to directly answer your question, not only were human rights impacted in Europe, but also in Latin America, based on the enlightened ideals found in the Declaration of the Rights of Man and the Citizen. The Haitian Revolution influenced leaders like Simon Bolivar to incite rebellions against Spanish rule in South American countries, so the larger impact of the French Revolution is the spread of democratic ideals and freedom for citizens (read: men) in the Americas.
One distinct way in which human rights were influenced by the French Revolution was through the Declaration of the Rights of Man and of the Citizen. This document played a vital role in constructing the theoretical underpinnings of the French Revolution. It sought to cast the French Revolution as a human struggle for rights more than a political one. The Document set the course for the appeal of the Revolution as a statement for all of humanity, not merely one group of individuals. The language in the document helped to articulate the condition of "human rights" in the modern setting. The language in the document helped to define the concept of human rights. Language such as "Men are born and remain free and equal in rights," "The aim of all political association is the preservation of the natural and imprescriptible rights of man," and "As all persons are held innocent until they shall have been declared guilty" are reflections of how the Declaration of the Rights of Man and of the Citizen influenced the construction of human rights.
The French Revolution's appeal to larger and collective notions of humanity can be seen in the document's wording. The French translation of the document is "Declaration of Human and Civil Rights." This speaks to how the French Revolution sought to influence human rights. It is in this light where the French Revolution believed it articulated a condition that was applicable to all humanity.
The French Revolution made massive waves in continental Europe. During the French Revolution and Napoleon's rule, the rest of Europe were freaking out, because the monarchs were worried that the revolutionary nature in France would travel to their countries and incite demands for reforms, such as freedom of speech and assembly and press. Karl von Metternich of Austria (he was the Habsburgs' Minister of Foreign Affairs) proposed to the other leaders in Europe about a plan for collective security, in which if one country has a rebellion, then the other countries will come in and help suppress it.
The French Revolution came from the philosophies of the Philosophes, the French philosophers/thinkers (say that five times fast :P). These people met in salons to discuss civil rights and reforms. France, prior to the Revolution, consisted of three estates, or classes. The First Estate were the Nobles, the Second were the Clergy, and the Third were everybody else. The First and Second Estates didn't have to pay taxes, and as France was heavily in debt, the Third Estate had to pick up the tab. This angered the poor, since they didn't have the money, and this angered the professionals (middle class) because they believed they deserved the same rights as the nobles (since many were actually wealthier than the nobles, but received fewer rights). The Third Estate during this time was demanding equal representation and many civil liberties.
After the French Revolution, despite Metternich's system, many nations, due to the rise in nationalism and demands for civil reforms and a constitution, had revolts. Some that succeeded were the Greeks (from the Ottoman Empire) and the Belgians (from the Netherlands). France continued to rebel whenever they were upset and demanded more rights, hence the July Revolution (1830) and the Revolution of 1848. In many European countries, a movement called Young Europe arose that inspired many insurgents to demand constitutional changes in the Austrian Empire, Hungary (Austria), Ireland (Britain), Poland, and many others.