Lots of phenomenon in nature can be plotted onto some sort of normal distribution, and that makes intuitive sense to me (such as human height). Why then, does global human life expectancy drop off so suddenly after ~85 years old? Is there some factor which suddenly increases mortality significantly? How come so few people (statistically speaking) age above 85 years old?
The histogram you show isn't a distribution of age-at-death for individuals, it's a distribution of life expectancies for different countries. The post you link takes data from a table on the Wikipedia page List of countries by life expectancy. Interpreting the histogram you posted, we see average life expectancy seems to hit a wall around 85. This does not mean that individual human life span is not normally distributed.
Excluding deaths at infancy, the peak of the death rate curve appears to be at about 88 years for women in the United States in 2014, as seen in a blog post by the The Max Planck Institute for Demographic Research:
There's no particular reason everything in nature needs to fit a normal distribution; normal distributions are just one of many different types of naturally-occurring distributions. In some cases, the other types can be transformed (e.g. by logarithms) into a normal distribution (e.g. a log-normal distribution), in other cases not so much.
As to why it doesn't match a normal distribution? Normal distributions tend to occur when they're the sum of many independent processes. Mortality factors aren't independent. Many factors that affect mortality are age-dependent (cancer, heart disease, etc), so they depend on the same factor (age) and accumulate at the same time, not independently. Therefore longevity doesn't meet the basic expectation for a normal distribution.
If you look at specific mortality factors, you might well find some that are relatively independent of age, and these might be closer to normally distributed. But age is extremely important for most forms of mortality, even those that don't accumulate with age (automobile mortality is higher in younger drivers; infectious diseases disproportionately target the very young and very old; maternal deaths are more common in younger to middle-aged women).
Instead, longevity is a negatively skewed distribution. It may crudely be considered as a log-normal distribution, though not perfectly. There are also more sophisticated approaches to model mortality distribution, such as the Gompertz–Makeham model (simplistically, an exponential increase in death rates with age), or the Weibull distribution.
Various publications compare models, or come up with new and shinier models, for longevity: