Really interesting book on the subject:
TLDR: 2 unchangeable sexes + death (both of those) has proven to be the most effective reproductive strategy, beating out every competitor by a huge margin. This means there is a very good reason for death to exist and death is a "new" thing. There was a very long time when animals (well ... technically multicellular bacteria colonies) just didn't ever die. Some of those are alive right now and are hundreds of thousands of years old, at least. We know about the effectiveness because there are species (none are animals) that are still alive that predate evolution's "invention" of death, that have competed to conquer the earth, and they failed dismally. And we know about the 2 unchangeable sexes because there's a lot of variation in sexes, none very successful (there even are asexual animals, and animals that can change their sex, and animals with different sex arrangements. Some have small differences (like males giving birth), and some are hugely different, like true asexual animals)
If you analyze death in humans you will quickly see it's not as simple as you might think. Cells die in humans all the time, in fact millions of your cells will have died before you finish reading this post. Some organs, such as the skin and the colon, depend on killing (ie. triggering programmed cell death) large amounts of their own cells to function, and even internally it is used as a tool where more gentle means would probably work, such as bone growth and repair, which involves large amounts of cells dying.
Second if you analyze what do humans die from if they die "peacefully in their sleep" ? Well they die of "natural causes". That's awfully noninformative. What do you die from, really ? Well, you die from energy exhaustion in your blood, causing multiple organ failure, leading to poisoning, leading to more organ failure, leading to more organ failure, leading to a relatively slow and orderly shutdown of your body. At this point there are certainly things that your body could do to stay alive (all it takes is accelerating energy dumping in the blood), but it doesn't. Your body chooses to die at a certain point, and it very gently kills itself at a certain point. There's a feedback mechanism that makes this happen. It takes something like a few weeks to actually die. People often report that they can feel this happening and the timeframe seems to be such that you have plenty of time to say goodbye or do whatever needs doing.
Because this is how death works in animals, it does mean that if someone dies from natural causes, the organs are thoroughly poisoned and have had to take extreme measures to avoid dieing completely before the body dies. This usually includes sacrificing significant parts of the organs. This means you cannot safely transplant organs from people who die slowly. But this is another subject.
Thirdly while everybody focuses on death, there are pretty serious symptoms that occur, frankly they start happening before you're even born, but by the time you're 60 there will be very obvious external symptoms of aging. Most of the ones we focus on boil down to cell senescence: your cells choose to systematically become less active as they age, and at a certain cell age (measured in generations) they kill themselves, and they are mostly, but not entirely replaced (that's the function of stem cells: replace senescenced cells that have killed themselves, and that's why they're such a big focus for anti-aging research). This is a mechanism that, essentially, stores your age, and the total energy use of the environment of the cell, into every DNA chain in every cell, and responds to it. Your age gets too big, and it kills itself. You use a certain amount of energy over time, and your cells kill themselves. Long before they kill themselves, they will force themselves to use less energy, even when it means things go wrong (this is why you should -and will- systematically eat less as you age).
And of course there are exceptions to death. There is a continuous cell line from every living human (even clones and the like) to the first human pair, and probably even to the very first living organism. So specific cell types, most notably the procreation related cells, are exempt from death. They do age, as in they measure their age, in years and energy, and they do kill themselves (even a lot quicker than normal cells) but they reset their age every time procreation happens. So, assuming you've got children, not all of your cells will die when you die.
There is a clear evolution to death, as a mechanism. Early lifeforms did not age (some are still alive, so we should say "do not age" in a few cases. Some were alive the first time Eve proverbially smiled at Adam and may still be alive when the last human dies). They would die from disease, getting eaten, or simply by getting themselves into a situation where they could not survive (getting buried was pretty popular). Of all of life's "kingdoms", there's 5.
* The Monera (mostly bacteria, mostly single cellular organisms, with many interesting exceptions), do not die of old age. A curious exception is that a number of them have a built-in self destruct (ie. death) if they do not reproduce.
* The Protoctista and Fungi (2 kingdoms). Have chromosomes and a cell nucleus. Some living protocta are hundreds of thousands, maybe even millions, of years old. They are mostly single cellular organisms, but some of them are very big, for example some algae can be 40 meters or longer. The general rule seems to be that sexually reproducing protoctista die, but there are many interesting mechanisms of death found in these cells, rather than just the one we see in higher lifeforms.
Many can "pause" their death clocks. Only time spent eating causes the death countdown to tick and various forms of hibernation do not.
Quite a few are "optionally" sexually reproducing and they disable and reset the death counter when they switch to the asexual mode.
Some have inherited death clocks: organisms can switch to asexual reproduction but their death clocks do not reset upon reproduction. If they, and their offspring, do not switch back before a hundred or so generations, they age, grow weak, and die.
Keep in mind that when single cellular organisms sexually reproduce, they start with 2 cells and end with 4 cells. Mostly, there is no difference between parent and child. They change their own DNA and then divide. So "parents" don't die sooner than their children, and mostly you simply cannot tell the difference between parent and child at all. And all of the cells have their death clocks reset.
Despite being single cellular organisms, they actually have sexual "organs" (parts of the cell dedicated to reproduction alone). Both outer and inner ones : a way to connect the cell membranes together, and dedicated "micronuclei" that actually reproduce. The cell then "kills" (eats-sort of) the "old" cell macronucleus. Generally the new cells immediately recreate their "sexual organs", which are deactivated immediately after creation, and the newly formed cells go about their business under the direction of their new macronucleus.
The "death" of the old nucleus is the first form of death that is encountered in evolution.
The defining feature of death seems to be that DNA is split into two parts: DNA that is used exclusively for reproduction and DNA that is used for, well, anything else. The reason you die is that a human being falls into that second category.
The obvious variant that must have existed yet is missing is the variant we all want to exist: sexual reproduction where the death clock is reset in both parent and child, and the parent DNA is not destroyed. We do not know of any lifeforms that do this.
So there is a simple and inescapable conclusion : sex (the sharing of genetic information of multiple individuals in a species) does not work without death. That is why you die.
Extremely interesting book on the subject: "Sex and the origin of death"