One way of looking at evolution is to see yourself, the individual, as little more than a disposable short-term delivery system. The focus of evolution is propagation of the germline, and aging exists in its present unpleasant form because in 99.9% of all complex species there is no selection benefit in avoiding it. On the one hand, nature is red in tooth and claw, and the only system that survives in the wild is one that gets the job of replication done before a violent or diseased death. On the other hand, systems optimized for early life tend to fall apart and consume themselves in later life. The mammalian adaptive immune system is a good example, a limited capacity system that will eventually malfunction due to encountering and attempting to remember too many different pathogens regardless of all of the other issues of aging. Evolution led to that system because it works well enough to get by in early life, and because there is little selection pressure to avoid the inevitable crash later in life, when the chances of reproductive success are low.
Lastly, there appears to be a race to the bottom between long-lived and short-lived species. The reason why we see so few species succeeding in their own niche via a long-lived strategy of agelessness and continual replication, as is the case for some species of hydra, may be that aging species can adapt more rapidly to changing environments. Thus highly regenerative, ageless species of various sorts may arise over and again in larger numbers during long periods of environmental stability - it is hard to say from the fossil record whether or not this is the case - but are out-competed and swept away by aging species when the climate or ecology shifts rapidly enough over evolutionary time.
Many people would tell you that death - or rather, aging - wasn't around until we started reproducing sexually. There's no reason sexual recombination in itself would demand our death, however. In fact it clearly doesn't: we know of two species of worm which reproduce by splitting themselves lengthwise and fusing together, who are nonetheless no likelier to die in old age than in youth. The famously immortal Hydra is also capable of reproducing sexually (although it usually chooses budding instead). On the flip side, there are multiple organisms who produce eggs asexually, but aren't any safer from senescence than we are.
The rather disturbing truth about life and death is that our bodies are just disposable vessels for the replicators cushioned safe and snug within our germ cells. While a few organisms like the Hydra kept things simple by remaining as one with their germline, others built free-standing bodies of somatic cells with ever more complex machinery to house and propagate a germline that was reincarnated each generation inside a new (and hopefully improved) body. The resulting collections of meat and bone eventually became complex enough to totally obscure the germline itself, and conducted lives with apparent independence - humans in particular enjoyed millennia of ignorance about our fundamental irrelevance. But despite its obscurity, the consequences of its influence could hardly escape notice, for one simple reason: once the germline had abandoned the body, we were all condemned to death.
Key to this fate was the fact that you would, at some point, probably die anyway. Maybe you'd starve in a famine, or be eaten by the resident apex predator, or just freeze to death. But whatever the cause, you were always less likely to live two years than to live one. In this way, each individual's reproductive potential was concentrated at the beginning of his life, and declined at some rate after sexual maturity. The most crucial task was getting you to reproductive age at all, and consequently a lot of selection power had to be spent on birth and development. By contrast, ages that organisms were rarely capable of reaching would experience extraordinarily little selective pressure, the bottom of a genetic slump that began very soon after reproductive maturity. The problem wasn't just that beneficial alleles in old age weren't selected for; it was that alleles with damaging effects later in life could curry favor by increasing fitness at the ages of highest reproductive potential - an effect known as "antagonistic pleiotropy".
Exactly how bad a deal you, the body, get is dependent on your niche. How dangerous is it? How long are you likely to survive? If you're a wild mouse, your chances are around 10% in the first year. There's no sense in spending a long time on growth and development, and you'd better have lots and lots of children, because most of them will die. On the opposite end of the spectrum are organisms that live in relative safety. They may live on an island with no predators, such as a particular strain of opossum with unusually long life, or have an unusually reliable food source. Humans, for our part, didn't come out as bad as we could have. We're among the longer-lived species, and there's a good chance that apparently unrelated medical advances in the last century have been pushing us in the direction of a slower intrinsic aging rate.