In this usage, short means:
5a : not coming up to a measure or requirement : insufficient in short supply
b : not reaching far enough the throw to first was short
c : enduring privation
d : insufficiently supplied short of cash, short on brains
Sense D is the most exact match; the idea is that a "normal person" has enough sense or capability to achieve a standard measurement (represented by some arbitrary item, in this case a package of beverage containers that normally has 6 units), but the person being talked about does not have enough sense or capability to achieve that same measurement.
"I was X short" is generally used as a quick way to say "I needed (some number N) of something, but I only had (some smaller number, Y)", then X = N - Y, so if you need $12 to buy something but you only have $11, you are one dollar short.
The expression, then, equates intelligence to beer: you need 6 units (cans) to be normal, but the person being discussed is "one short of a six-pack": they have only 5 units of intelligence, not the 6 expected.