[Mashable advice] It’s superfluous most of the time. Open any document you’ve got drafted on your desktop, and find a sentence with “that” in it. Read it out loud. Now read it again without “that.” If the sentence works without it, delete it.
The idea is fraught with problems, most turning on the fact that there are several distinct lexical items that, with a large number of uses, and with distinct syntax, discourse functions, and sociolinguistic statuses for each use.
Part 2: On one of these items, the complementizer that, and its use to mark the object complement of a verb, as in
They know (that) pigs can’t fly.
(where the that variant and the ∅ variant are both fine).
But then I started an e-mail to a friend:
I do wish people would credit sources.
(with the ∅ variant; the that variant is also possible) and thought to link to previous context with a though — but then the ∅ variant struck me as very awkward indeed:
?? I do wish, though, people would credit sources.
though the that variant is fine:
I do wish, though, that people would credit sources.
What’s crucial is that material intervenes between the complement-taking verb and the complement. It turns out that this intervention effect is well-known in the variation literature.