What is the origin of the rule for omitting the suffix of a hyphenated word?

When it uses a hyphen specifically, it's a suspended hyphen.

This type of thing isn't easy to search for, but I found an example that uses it as far back as 1826:

Upper- and under-jaws (mandibulæ and maxillæ).
_{An introduction to entomology; or, Elements of the natural history of insects (via the OED)}

There are earlier examples where the first element doesn't have a hyphen, such as this one from 1818:

Bring the high and low-lights upon the Fern in one, proceeding in that line.
_{New & Extensive Sailing Direct. for Navigation North Sea}

It's a variety of Conjunction Reduction, used to avoid repeating material that's already been said. In this case, it's morphological instead of syntactic, but it's got the same purpose and works much the same way.

Generally, though, once it's been reduced, there is a preferred order for such oppositional phrases as high- or low-budget; low- or high-budget sounds strange. And they can't get too far from one another, either:

• *He left the low-, but she picked the high-hanging fruit.

is terrible.