The standard answer to this question is protection: human rights offer people protection against other people or the state. People need rights because they want to protect their interests, their freedom, their equal status, their opportunities, their values and their projects in life against attacks by those more powerful. (There’s a more elaborate version of this standard thesis here).
However, there’s a sense in which we need rights even if no one harms anyone else. Immanuel Kant has made this claim in a very convincing manner. Suppose that all those people who are powerful and strong enough to frustrate our interests, projects, opportunities and values and to harm our freedom, independence and equal status refrain from doing so in a coherent, systematic and predictable manner. Hence, there is no harm imposed by people on each other, and one could assume that human rights retreat to the background. In fact, they would seem to become totally useless.
And yet, such a social setting would imply that the weak are able to enjoy their rights, their freedom and their equal status and to pursue their goals and values only “on the sufferance of the strong” and with their explicit or implicit permission and indulgence. Kant thinks, rightly I believe, that it is wrong for people to be dependent on others for their freedom and equality in this manner. And if we understand the “weak” to be almost everyone – even the strong have to sleep – then this dependence on indulgence will be a general phenomenon. Hence, even in such a seemingly idyllic society awash with benevolent and self-restrained power we need human rights.