Unit tests as documentation

One - unit tests explain nothing. They show what the output should be for a given input, but not why, or how you get there. I'm surprised by the nonchalant claim that "unit tests explain code". Am I missing something about the meaning of the english word "explain"?

Two - so any input value outside of those in unit tests is undocumented / unspecified behavior? A documentation can contain an explanation in words, like what relation should hold between the inputs and outputs in all cases. Unit tests by their nature can only enumerate a finite number of cases.

This seems like such an obviously not great idea...

One: Can we test the tests using some somewhat formal specification of the why?

Two: my intuition says that exhaustively specifying the intended input output pairs would only hold marginal utility compared to testing a few well selected input output pairs. It's more like attaching the corners of a sheet to the wall than gluing the whole sheet to the wall. And glue is potentially harder to remove. The sheet is n-dimensional though.

I really don't understand the "exhaustive specification" thing. How else is software supposed to work but with exhaustive specification? Is the operator + not specified exhaustively? Does your intuition tell you it is enough to give some pairs of numbers and their sums, and no need for some words that explain + computes the algebraic sum of its operands? There are an infinite number of functions of two arguments that pass through a finite number of specified points. Without the words saying what + does, it could literally do anything outside the test cases.

Of course, for + it's relatively easy to intuit what it is supposed to mean. But if I develop a "joe's interpolation operator", you think you'll understand it well enough from 5-10 unit tests, and actually giving you the formula would add nothing? Again I find myself wondering if I'm missing some english knowledge...

Can you imagine understanding the Windows API from nothing but unit tests? I really cannot. No text to explain the concepts of process, memory protection, file system? There is absolutely no way I would get it.