-The purpose of morality is to allow a society to function.
-There are many different sets of moral beliefs.
-Some sets of moral beliefs are better than others.
-The value of a set of moral beliefs can be judged according to how well it serves its purpose.
-The best set of moral beliefs will lead to the best functioning society.
Based on the above principles, we can objectively observe the value of our moral beliefs, and compare them to other cultures, as well as the beliefs of the past. If our moral beliefs today are giving us a better society than we had a thousand years ago, then we can say we have observed an improvement in our morals. That's how we can empirically observe morality.
We can also use an inductive approach to the issue. The purpose of morality is to allow a society to function. If we are to look at two societies with two different moral codes, we can make logical guesses about which society will be more prosperous. For example, if society A does not allow women to speak in public, and society B grants full equal rights to women, we can logically deduce that society B will have more female professionals who contribute to the society's prosperity than society A, and will therefore be more prosperous.
Using a combination of inductive reasoning and empirical observation, we are able to continually progress our understanding of morality towards objective moral truth.
That's how atheists are able to understand natural rights. The progress that our culture has already made in this direction has steered us towards a belief that when natural rights are enforced, societies prosper. When they are ignored, societies fail.
No God required.
I'd like to also add that religion in general has acted throughout history as a damper on our moral progress, by means of its prohibition on moral inquiry.