Samuel Moyn wrote a book called The Last Utopia: Human Rights in History. At base his argument is that the concept of universal human rights is a reaction in the last say 50 years against the idea that national sovereignty is a path to the guarantee of rights.

I think his argument goes a bit too far in dismissing times when universality had come up in previous discourses on rights, but I do think his analysis, at least, of how the concept has developed over the past 50 years to be fairly useful. It has become a moral, not political, notion about rights. Therefore, in some ways human rights are a reaction to the failure to provide civil rights and in fact goes against the idea of national self-determination as the cure all(which still has a lot of power itself, just look at the discourse surrounding the recent Arab Spring). Projects even like the American and French Revolution, he argues, which we now often retroactively associate with the idea of human rights because the words are sometimes strikingly similar, in fact belong to a history of very politicized rights distinct from the depolitical, almost anti-political, notion of human rights today.

This is notable because national self-determination was supported in part as a cure for imperialism and colonialism. What became obvious is that it wasn't the panacea people had hoped. The modern concept of human rights, Moyn argues, is an attempt at universalism through the concept of rights. The idea of universalism is important here because is is clear that there have been other proposed universalisms in history. Take, for example, the notion that many religions have that humans are united under god. Moyn says a universalism based on Human Rights is distinct, if part of the same lineage as these earlier universalisms.

So what changed?

Moyn quotes Bronislaw Baczko as saying "It is as if utopia were the scapegoat in a collective exorcism of the misnamed and ill-defined demons which haunt our epoch... Is it not possible that the disenchantment with utopian 'systems' goes together with the persistence of diffuse utopian hopes and modes of thought which together might betray the presence in our times of two contradictory attitudes: distrust of utopia with the desire to have one anyway." (Moyn, 120)

In other words, utopian projects in the 20th century generally ended in disaster, and recognizing this fact - the idea of utopia, especially revolutionary political utopia I would argue, but the general idea as well is not trusted anymore. Nonetheless, we seek a path towards one. Moyn argues that the modern discourse on human rights became our way to have our cake and eat it too. On the one hand it remains a utopian ideal itself, on the other hand it rejects most of the baggage of previous utopias. Therefore, Moyn says "Human Rights emerged as a minimalist, hardy utopia that could survive in a harsh climate." (121)

So this explains (if you believe Moyn) the presence of the discourse of Human Rights over roughly the period about which you are asking, but it doesn't necessarily explain why they've "escalated." First of all, I wouldn't overstate just how much "progress" we've made in this regard. Things like the civil rights movement in the United States certainly made strides in terms of legal rights, but there are still a lot of places in the world where legal differences remain. And it isn't a stretch to say that even in the presence of de jure equality de facto inequality continues to exist. But this is still very much an ongoing struggle. Perhaps historians in a couple hundred years will have something more to say.

Many people have successfully traced back the idea of human and civil rights to, at the very least, the high/late Middle Ages, and have interpreted various scholastic thinkers (notably Thomas Aquinas, and its interpreters of the school of Salamanca) as precursors to the idea of human rights. Not to say, of course, that these people were anti-colonial and feminist activists in disguise; but there is in my opinion a strong case to say that Christian thinking and philosophy carried the seeds that would later produce the idea of human rights. I do not resist quoting a few lines from Agobard of Lyon, bishop in the 9th century:

There is now neither Gentile nor Jew, Scythian nor Aquitanian, nor Lombard, nor Burgundian, nor Alaman, nor bond [slave], nor free.

A very interesting text from the same period was written by a certain Ratramnus, monk in Corbie, who argues in a letter that Cynocephali (dog-headed men) can be considered as men and not as animals because they live in communities, a very aristotelian argument that led him to an astonishingly universalist conclusion. To be honest, he also relied on something that we would have more trouble understanding nowadays, i.e. the fact that these purported Cynocephali were cattle raisers, a privilege of men according the usual interpretation of Gen 2:20. Similarly, Agobard relied on a religious conception to make his aforementioned statement — but the equality of men before their maker was still the theoretical basis of John Locke. On a more practical level, Christianisation also created actual rights — while the decline of slavery seems to be due to economic factors rather than Christian norms, the end of divorce can actually be considered a progressive idea. At that time, divorce was almost always repudiation; its end was a protection for women rather than an enslavement.

I would love to elaborate more on this, but it is reasonably clear that Christianisation played an important role in the appearance of the mental frame that made the concept of human rights possible. Tocqueville has written, as always, very interesting analyses on the rise of human rights; he links them with an increased capacity of empathy, precisely something that has been a central part of the Church's teaching (he underlines, in a famous passage, the fact that Mme de Sévigné (a 17th century writer) does not see any problem in undressing in front of her valets; precisely, he says, because she did not consider them as fully-fledged human beings). As far as I know, an overarching treatment of this very complex problem still remains to be done (but it would lead an incredible amount of knowledge on 2000 years of cultural history).

I'm not a historian but I do specialize in International Law and Human Rights.

I think first of all, human rights isn't a "new" phenomenom. The déclaration des Droits de l'homme et du citoyen already had quite a few essential human rights enshrined and was made in 1789. Looking at other countries we can find quite a few countries with human rights in their constitution around that same period. So the premise that human rights is a new creation and that people suddenly wanted more the past 50-100 years is not really accurate. Human rights was just more or less a domestic issue, legally speaking.

Now the reason why only in the last 50-60 years human rights has been elevated to a more universal character can be found in, as said earlier by TenMinuteHistory, the diminishing of the absolute character of state sovereignty. This has always been one of the key discussion points in international law: state sovereignty versus the need for a "trancendental" set of laws. The establishment of the UN ( yes the league of nations existed before that, of course, but pales in comparison with the UN, from a legal point of view ), created a better forum to actually create international law and has greatly contributed to its development.

Now, human rights themselves can also be categorised or be seen as waves ( although there is quite a bit of discussion surrounding this type of categorising in generations ).

The first generation are the basic human rights: political and civil laws.

The second generation are the social, economical and cultural rights.

The third generation is, more generally, the right to development which is still being created and contested. The second and third category are in fact more "recent" creations in the field of international law.

And even now in this third generation there are still numerous issues surrounding state sovereignty. Simple example: when country A has a domestic economic policy that negatively impacts a poor population in country B ( for example certain farming subsidies ) this would, generally speaking, not be a violation of international law in the traditional sense. Yet, the right to development in extreme cases might argue that any negative impact caused by a domestic policy is a breach of international law, the right to development, and should thus be penalized. In this we can clearly find the argument of state sovereignty versus the transcendental character of international law.

If necessary I can give you a few sources on international law in general and more specifically the right to development and its discussion surrounding state sovereignty.

One big change that's happened was the Industrial Revolution. Prior to the IR, a huge determinant of your status in society was ownership of land. If you had land, you were somebody. If you didn't, you were much less important. However you could still be valuable if you were physically strong, and could perform hard physical labor or swing a sword. If you were busy being pregnant or nursing small children for a significant portion of your adult life, you were out of luck. With the coming of the IR, all that changed. Now almost everyone could find a job and support themselves. Most jobs didn't require great physical strength, which meant that women and children could now do them. Advances in medicine allowed women to have fewer children without worrying that most of them will never see adulthood because of disease. Now even marginalize members of society could achieve financial independence. In short, there was a significant increase in social mobility. Anyone could now become anybody. You were no longer limited by your birth or what you inherited.

Now it should be said that the Industrial Revolution was simply one of the vehicles through which equality and the leveling of the playing field were achieved. Philosophical foundations of the idea that all men and women are equal predated the IR. However even though such ideas existed, there were fairly limited means of turning those ideas into reality. Secondly, the benefits of the IR didn't always immediately translate into actual civil rights. For example, women's suffrage wasn't achieved until significantly after the IR. It took a long time for people's attitudes to change.

In the last 50 years I'd say news, particularly photojournalism, really opened people's eyes to human and civil rights abuses. Photographs such as the body of Emmett Till (which is NSFW/NSFL thus I am not providing a link), this picture of a police officer beating an African-American,or these firemen turning a firehose on civil rights protestors began to evoke sympathy for African Americans in a way that words alone could not achieve.