By which I mean, did they believe (abstractly at least) that their sacrifice was achieving something, or would the reasons for it be alien to them? Clearly this depends on where they came from, so I guess part of that is how dominant was the Aztec cultural-religious belief behind the sacrifices when it came to neighboring and vassal states?
Additionally, assuming that they did believe that there was a purpose, what would they have understood to happen to them after the deed? Was this some noble death on their part that would see some sort of reward in the afterlife, or what?
I can't speak to how people who did not share similar beliefs with the Aztecs would understand their sacrifice, but I can pass on some information of how those who did share those beliefs saw their fate.
This comes from The Aztecs (Peoples of America) by Michael E. Smith (bolding mine):
The Aztecs believed in several distinct afterworlds, and one's fate depended upon one's status in life and upon the manner of one's death. Soldiers who died in battle and sacrificial victims went to an eastern solar realm to accompany the sun during its rise to zenith. Women who died in childbirth went to a western solar realm where they accompanied the sun during its setting. People who died by drowning or other causes related to the rain god (such as lightning or certain diseases) went to the earthly paradise of Tlalocan. Most people, however went to one of the nine levels of Mictlan, the underground realm of death.
So to sum up, sacrificial victims would go to an eastern realm just like warriors. I think I remember reading that certain sacrificial victims may have gone to a different realm depending on how/to which god they were sacrificed, but I'm not certain on that. Could anyone elaborate on that point?
Also, keep in mind that sacrificial victims would not only be captured warriors. Depending on the god they were being sacrificed to specific types of people were required for the correct form of the ceremony. Here's a bit more from Smith's book:
The victims of this ritual were not considered ordinary mortals. They were viewed as deities whose deaths repeated the original sacrificial deaths of gods described in myth. The key Aztec concept here was ixiptla, often translated as "deity impersonator."
...
Most gods required warriors for their ixiptla, either purchased as slaves or the secondary offspring of nobles. Women were sometimes sacrificed as ixiptla for female deities. The most stringent requirements were those for Tezcatlipoca for the sacrifice in the ceremony of Toxcatl. His ixiptla, selected a full year in advance, had to be a handsome, well-bred youth with no bodily imperfections.
So while captured warriors were important for sacrifice, they did not compose the entirety of sacrificial victims. Those non-warriors who were chosen from within Aztec society would probably have shared the belief system of those carrying out the sacrifice. The above all comes from amateur reading of Aztec history, so if anyone has critiques or suggestions for improvement please let me know.