Death makes sense because organisms and the cells that they contain may only live for so long. Your body's cells are forever dividing, because you grow, and you also lose cells in shedding skin/digesting food/bleeding and so on. In order to get more cells, the existing cells undergo a process called mitosis, wherein they replicate. In this process, the chromosomes in your cells cross over to give the new cells genetic information. However, each time that this happens, the telomeres on the chromosomes break off a little bit. At some point, they are just too short to reproduce anymore, and your body runs out of good cells. This causes problems like organ failure and many diseases... and then you die. That is, of course, if you don't get hit by a bus or contract meningitis or get murdered. There are a ton of ways that you could die.
Anyway, things must die. There are some six billion people on earth right now... can you imagine if none of us ever died? The earth would become far more crowded, and more people would suffer because there'd be more struggle for resources that are already poorly distributed. If no one died, the soil would be exceptionally poor, as decomposing biological matter is what adds nutrients to the soil and allows things to grow and thrive. The poor soil would make it nearly impossible to grow anything substantial for food (and shelter... recall how much we rely on wood) so we'd be screwed. What's more, if nothing died, we couldn't kill plants or animals to eat them. Everyone would starve, but since they couldn't die, would exist in a state of perpetual agony.
That's what makes sense about death to me: elementary biology. What doesn't make sense about that to you?