One of the hallmarks of the atheist (typically) is that he (she) trusts her (his) ability to reason more than the revelation of some other human being.
This tendency carries over into how we deal with death. The first thing I realize about death is that it is far from monolithic. Death is such a common phenomenon that we encounter it in many ways. The venue in which we encounter death determines how we deal with it.
We tend to (at least most of us) regard human death differently than non-human death. Then we regard the death of people we know (or feel we know) differently than those we do not know, and more differently still than "statistical" deaths where we do not even have a name to go by and no identifiable concept of the person(s) involved. That is origin of the idea that "one death is a tragedy. A million are a mere statistic." That is why the media loves murder stories–depending on the story (real or not) of the victim, a story of a murder where we are given a portrait of the victim is compelling.
This is almost assuredly a product of evolution. There would have been tremendous evolutionary advantage to a group where mourning the loss of the dead members of the group (or conversely celebrating the death of enemies or threats.) Such feelings would lead to greater group cohesion. The feelings we have when someone close to us (and to a lesser extent someone like us) dies are a product of evolution, chemical reactions brought about because nature selected for them.
Then finally there is the matter of our own deaths. Again, there would be an evolutionary advantage to a creature that became aware of the possibility of death and took actions to avoid that circumstance. That advantage would likely be encoded as fear.
As an atheist, I realize that my fears regarding death are:
2) and at least partially counterproductive.
Regarding that second matter, again, we have evolved to the point where at least in some cases our reason allows us to override our emotions (including fear.) I deal with the idea of my own death by realizing two things:
1) I did not exist for 13.7 billion years prior to my conception and birth, and that was not a problem for me.
2) Similarly, once I have become dead, I will no longer have a problem with death, as I will not exist and thus will not experience anything, let alone fear.
In short, being dead is only a problem while I am alive.
I honor and mourn the dead because it is emotionally healthy for me to do so, in the same way it is healthy for me not to eat processed sugars and grains. Evolution made me that way. The "spirituality" I feel when I mourn the dead is simply powerful emotion. Since I typically do not feel those emotions on an everyday basis, they seem somehow set apart and different. I realize they are only different in character and depth, but that they operate on the same chemical pathways as other emotions.
Hope that helps.