| I don't think there is just one answer, but rather a combination of things:
1. The point about metabolism rates is a good one, and small animals tend to have a higher metabolism (because the surface area of their skin is greater in proportion to their body mass) - but this really only applies to warm-blooded animals.
2. Thanks to human society, humans rarely die from predators, exposure, or starvation. Hence many humans have the good fortune to live long and die from "old age" whereas most animals aren't so fortunate. And when they get sick or are injured humans have the benefit of medical care to further extend their life.
3. From an evolutionary view point, think of the advantages and disadvantages of a species living a long life. For many species, having old members living for a long period of time after they have lost the ability to reproduce can be detrimental to the propagation of the species, as they would compete for scarce resources against those who are still of child-bearing age. Consequently the species is more likely to succeed if the old ones are "programmed" to age quickly and die off soon after they have stopped having offspring. But for many species that are quite social - such as humans, chimps, and elephants - it can be beneficial to the group to have old ones around, as they can help the group hunt for food, protect the child-bearing members of the group from predators and can assist with the burden of raising youngsters, thus allowing the parents to have more successful offspring. For these groups if your grandparents are alive it improves the chances of you surviving long enough to have offspring of your own.
There are probably other factors as well. | 
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