RE: Why looking-at-p works?

[Drew Adams]

> Where is it documented?  I could find it neither in the Elisp
> Reference nor in let's docstring.

You might need to read a tiny bit between the lines, but this
(from (elisp) `Local Variables') pretty much suggests it:

  Sometimes it is useful to give a variable a "local value"-a
  value that takes effect only within a certain part of a Lisp
  program.

It might have said "is in effect" instead of "takes effect".

  To take another example, the 'let' special form explicitly
  establishes local bindings for specific variables, which
  take effect within the body of the 'let' form.

Again, "are in effect only" instead of "take effect".

(Consider suggesting a doc improvement for this point -
`M-x report-emacs-bug'.  And yes, probably the doc string
should mention "_local_ variable".)