In antiquity it was believed that there was a "natural law" which imposed obligations upon the individual (such as being a good and contributing member of society, defending the homeland, and such).
The concept of natural (or inalienable, or self-evident, or human) rights (as opposed to obligations) arose during the Enlightenment as a means of challenging the concept of the Divine Right of Kings, which essentially held that kings were bound by no earthly authority and could, consequently, do whatever they wanted and treat people (their subjects) however they wanted.
Natural rights, the most fundamental human rights such as life, liberty, and the pursuit of happiness (or property, depending upon which point in the developing conversation on natural rights you happen to jump in on), it was proposed, were God-given to people as a natural consequence of being people, as opposed to being something that could be given or taken away by other people.
As such these rights couldn't be usurped by a monarch/sovereign who, while not bound by any earthly law, was subject to the laws of God.
When we (here in America but later in Europe and then all over the world) began to transition away from monarchies we maintained the concept of God-given natural rights.
Basically natural rights are rights because we, the people, say they are.
The concept of natural rights is a convention, nothing more, nothing less.
Why do we have a "right to live"?
Because collectively we say that we have a right to live.
Though the concept originally depended upon an appeal to God (because it developed at a time in human history when everything was almost universally believed to be dependent upon God) these rights are acknowledged universally today in placed both where the authority of God is held sacrosanct and in secular societies such as our own.