The idea of rights barely existed a thousand years ago, at least in the English-speaking world. The (Charter of Liberties)[https://en.wikipedia.org/wiki/Charter_of_Liberties] (1100 CE), precursor to the Magna Carta, was one of the first documents to specify a right to someone other than a king. If you want to stump your friends, have them guess what that right was. (It was the right of childless widows to remarry someone of their own choosing).
For the next several hundred years, the word 'right' primarily was used to draw a distinction between the rights of kings and the rights of (English)men. In French the word 'droit' can be translated as 'right' or 'law'. So it originally came into English as ideas about what kings can do (laws) and then became ideas about what king's can't do (rights). The thing about rights is that they are always, at root, claims about what government should or should not do.
That's the work the word is doing in the Declaration of Independence: sure the king has some rights, but here's all the rights the people of the colonies have that the king is trampling over, which he shouldn't have done. And likewise the Bill of Rights in the Constitution lays out what government can't and/or must do. Then the French Revolution happened, with le Déclaration des droits de l'homme et du citoyen de 1789 setting out the rights of French people. Now that literally translates to the "Declaration of the rights of man and of the citizen", but Thomas Paine and Mary Wollestonecraft both translated it as, 'human rights'. Mary Wollestone did so in a book titled, A Vindication of the Rights of Man (1790), but then circled back two years later with A Vindication of the Rights of Women, and more or less invented feminism.
Except for the elimination of slavery from most of the English-speaking world, surprisingly little happened in terms of 'human rights' until WWII. After the war, the Europeans and Eleanor Roosevelt decided the problem in Germany was that the other countries let Hitler steamroller the rights of all his citizens, and then he tried to do the same thing all over Europe. Winston Churchill in particular gave speeches arguing that if European countries were forced to observe human rights, then no despots like Hitler would ever rise again. He wanted Europe to be like the United States, although he wanted the U.K. to not really be a part of it.
Eleanor Roosevelt meanwhile was very influential in the founding of the United Nations. The express rationale of the United Nations, in it's charter, is to "reaffirm faith in fundamental human rights, in the dignity and worth of the human person, in the equal rights of men and women and of nations large and small". This leads to the U.N. Declaration of Human Rights, which makes human rights a global thing.
As this was happening, Black people in America realized that the U.S. was out there promoting 'human rights', but not practicing them at home. They weaponize rights language to great effect, forcing the U.S. government to acknowledge those rights at home, lest we look like giant hypocrites to the rest of the world. Unfortunately, a lot of people in the U.S. -- segregationists and other white supremacists -- felt that if Black people could have human rights, too, then human rights must be a bad thing. So after the civil rights era there was a lot less enthusiasm in the U.S. government for talk of human rights.
Elsewhere in the world, colonized peoples seized on the U.N. Declaration of Human Rights as justification for their liberation. Rights language played an important role in decolonization rhetoric -- as well it should. Here Churchill's enthusiasm for rights bit him in the ass: he was a huge racist who really believed rights applied to white humans, not all those other humans. But apply they did, and the British empire mostly crumbled as brown and black people asserted their rights.
In the U.S. and elsewhere, people who witnessed the demise of segregation and the success of liberation through struggle armed with rights rhetoric quite naturally decided to apply that strategy to their own liberation. At the same time, the increasing diversity of society has allowed all sorts of new identities to cohere that were not really thinkable in the English-speaking world until now. People who have always been a part of society now could self-identify freely, and so better organize towards realization of their rights. So gay rights (which has been ongoing for nearly a hundred years, really), trans rights, disability rights, etc. etc. etc.