Smaller animals generally have shorter lives because of the difference in the surface area to volume ratio between smaller animals and larger animals.
This difference in surface area to volume affects the amount of the total internal volume of an organism that is exposed to the surface. If that organism is a warm blooded mammal, having more internal volume exposed to the suface will decrease that organism’s efficiency at retaining the heat required for maintaining all the physiologic processes in the body. To compensate for this heat loss, the smaller mammal will have adapted to having a faster metabolism and heart rate to make up for and provide the heat that is being lost to the atmosphere at a faster rate. For example, the resting heart rate of the Etruscan shrew is 800 beats per minute and has been recorded as high as 1511 b/min. In comparison, the resting heart rate of an African elephant, is 25–35 beats per minute.
The compensating organism will also have to consume a much greater percentage of food relative to body weight, as compared to their larger counterparts to support this dramatic increase in metabolism. The Etruscan shrew, needs to consume twice its weight in food each day to support its metabolism and survive.
In contrast, an African elephant which is the largest land mammal, has a much lower surface area to volume ratio, meaning less of its internal volume is exposed to the suface and thus an African elephant is much more efficient at conserving body heat. The African elephant eats approximately 5% of its body weight in food daily. African elephants are so efficient at retaining body heat, they have even adapted large ears to radiate heat away from their body through a network of blood vessels contained there.
The African elephant’s average life span is between 60 and 70 years in the wild, while the Estrucan shrew’s is between 2 to 3 years.
One of the draw backs to working at an excellerated pace to keep a metabolism up in order to counteract heat loss in smaller manmals is the accumulation of wear and tear on the physiological system leading to a shortened life span for smaller creatures.
To repair the wear and tear that accumulates more quickly due to living at a faster pace, cellular division must occur more often. Each time celluar division occurs, the telomeric end caps that are responsible for keeping the dna from fraying, unraveling and becoming disorganized, damaged and unusable, shortens, effectively acting as a limiting factor for the amount of times cellular division containing any genetic material may occur effectively.
So why do smaller animals live shorter lives?
Because of the wear and tear of an organism as well as the subsequent increased rate of cellular division accompanied by the shortening that occurs to the telomeres of dna after cellular replication due to living at an accelerated metabolic pace in response to heat loss due to the surface area to volume ratio that disfavors the physiology of smaller creatures when it comes to retaining heat in an environment.

Ok, so let’s get this out of the way: animals don't have unusually short lives,  we  have unusually long lives. Also, like other people pointed out, that chart is bullshit.
Yeah, it’s related to body size and all, but our closest evolutionary cousin, the chimpanzee, only lives to be around in their late 40s in captivity, and same for gorillas, which are much larger than us, meaning that in theory they should outlive us. But they don’t.
Our average lifespan in “captivity”, meaning with modern medicine, good nutrition, no predators and all that good stuff, is 80–85 years on average, with 100 years being the maximum for 90% of people. This age range corresponds to organisms weighting literally weighing a couple metric ton or more, so what gives?
The answer, as others pointed out, is genetics.  User-12184640813256344346  pointed out the function of the PEPCK-C gene, which may influence metabolism in mammals. A change in the regulatory section of this gene in humans could prove to be the difference between them and us.
I would like to point out another gene, the APoE gene. This gene is found unaltered in all apes, but in humans there are three exclusive variants of the gene: APoE2, APoE3, and APoE4. No human alive today contains the unmodified APoE gene as found in apes.
These genes mainly regulate blood cholesterol and lipid metabolism, which explain why they came about when early hominids started eating meat and other animal products, as they are rich in fat and cholesterol. However, this gene also has been found to play a role on degenerative diseases such as Alzheimer's, which are one of the main causes of natural death in humans.
This gene variant, especially the E2 and E3 forms, gives humans better protection against heart disease caused by cholesterol, which apes in general are very prone to regardless of diet, and against degenerative diseases, which explains the big jump in lifespan, both average and maximum.
This also explains why the weaker version, E4, is the largest standalone genetic factor in determining the chance of getting Alzheimer's, and why they have on average 6 year shorter lifespans than the others. They have also been associated with higher rates of atherosclerosis and their blood cholesterol and triglycerides are higher than the E3 and E2 counterparts. The only advantage to this mutation was higher blood levels of vitamin D, which may explain why this gene just didn't die off.