Human rights are not like music or love, things people care about for their own sake or for the pleasure or happiness they provide. If people care about human rights they do so because they view those rights as means to achieve some other goals or purpose. I personally see the following reasons why people care about human rights.
Human rights function as signaling tools (see also here). People who engage in human rights talk don’t necessarily have as a first priority the goal of improving respect for human rights, but perhaps only want to convey some meaningful information about themselves and use human rights talk to do that. They may of course improve respect for human rights along the way, sometimes unwittingly (for example because their talk contributes to a culture of human rights), but human rights are valuable to them primarily because they allow them to communicate certain things about themselves. For example, it’s possible that some of the people who are very expressive about perceived discrimination of a particular minority group may be primarily motivated by a possible leadership position within that minority group. Their human rights talk signals leadership aspirations. This kind of reason to care about human rights is not by definition useless for the promotion of human rights – it can advance the cause of human rights – but it’s obviously not the best possible reason.
Human rights promote people’s self-interest. That’s obviously true for their own rights, but also for the rights of others. I’ve written here about the ways in which people may view the promotion of the rights of others as a means to protect their own self-interest. This reason to care about human rights is more beneficial to the cause of human rights than the signaling reason, but it’s still not the best possible reason. People’s self-interest does not advance all human rights of all other people.
People may believe that human rights promote some of their cherished values or ideals, such as freedom or equality, for themselves and for humanity in general. Like the previous two reasons for caring about human rights, this reason will only advance the cause of human rights contingently: if freedom is what you care about, you will only promote human rights to the extent that they enhance freedom, and only those rights that enhance freedom.
It’s not true that freedom is served by all human rights all of the time, at least not if you adopt a restrictive definition of freedom. For example, those human rights that guarantee a basic standard of living are not clearly meant to enhance the freedom of poor people – within the bounds of a certain definition of freedom – and they may even limit the freedom of those who have to contribute the means necessary to guarantee a basic standard of living for others. Conversely, if you’re an egalitarian and equality rather than freedom is what you care about, then you may not feel especially attached to the right to property for example.
People may cherish rights because they believe human beings are uniquely valuable creatures who should be treated in a certain way. For example, you may believe that people deserve to be treated with dignity and respect. Or that they deserve to have their autonomy protected. Or that people have been created by God in His image and that this requires a certain treatment. Human rights are then viewed as means to achieve this treatment. Compared to the previous reasons to care about human rights, this reason is potentially more inclusive and wide-ranging and less contingent on facts about personal motivation. However, it depends on a substantive and inherently controversial philosophy about human nature, dignity or religion.
Conversely, people may cherish human rights, not because of their views about the inherent worth of human beings, but because of the evil inherent in humanity. Human rights are necessary not because they protect the good in people but because they protect people from the evil in others.