What does military plane’s escort (of an American Airlines flight) was “out of abundance of caution”’mean?

It means that even though it may not have been necessary, the officials sent the escort as a precaution. "Caution" is a synonym for "carefulness" and "prudence," so in acting out of an "abundance of caution," the officials were acting with a lot of carefulness.

The common English saying "Better to be safe than sorry" expresses the same idea.

The origin is a Latin phrase, ex abundante cautela, used in Roman courts, and transferred to English/American courtrooms, and then to everyday life. The original phrase refers to verbiage in a contract specifying (for example) what happens if a 90-year old has further children. Since there has been a recent move from using Latin phrases which not everyone understands to using English translations, which (as in this case) not even the user understands, the usual meaning is approximately "there's no good reason".

It means, "somebody panicked and we wasted $50 thou of the taxpayers' money almost shooting down a civilian airliner and now we need a catch-phrase to justify it."

"Three people locked themselves in the bathroom." Everyone locks himself in the bathroom. On an airplane, the light won't even go on until you lock yourself in there. The (false) implication was, people locked themselves in and wouldn't come out.

In fact, two people spent more time in the bathroom than some third person felt was appropriate. When the two finished using the bathroom for its intended purpose, they returned peaceably to their seats.

The upshot was people sitting near the third person gave him that "Dude, what is wrong with you?" look and the whole matter was forgotten. Hah-hah, I'm kidding of course: the upshot was the US Air Force (old motto: "Aim High!", new motto: "Faster than you can take a crap!") scrambled two F-16s in order to ...

Well, what? In the four serious attempts to crash US airliners since the WTC attacks, the attackers in every case were tackled and immobilized by fellow passengers, who reasoned, correctly, that it wouldn't matter to them whether they died crashing into a corn-field or into an office building but it make a big difference to other people. Even if those two guys in the lavatories had been carrying something more dangerous than loose bowels, they still wouldn't have been able to commandeer or destroy the plane.

The F-16s, by contrast, were perfectly capable of bring down the flight and killing all aboard. Don't think it could happen? Ask the passengers of Korean Air Lines Flight 007 or Iran Air Flight 655. They too were brought down by "an abundance of caution."

OBLIGATORY ENGLISH.STACKEXCHANGE CONTENT

Why do I bring this up here? Why not post it in rants.stackexchange.com? (No, it doesn't exist but I totally fell for it the first time somebody suggested I go there.) Because you don't have be George Orwell to see that corrupted language is used by dishonest people -- dishonest politicians in particular but dishonest people in general -- to hide what they're doing and thereby continue doing it.

Yes, "an abundance of caution" literally means something like "better safe than sorry" but in this context and many others it means "I panicked and wasted a lot of money and endangered some innocent people and now I want you to help us all pretend I'm not an idiot".

It's important that we use language (when we speak and when we listen) precisely and carefully, so idiots like the FAA guy don't get away with their idiocy.

I now return you to your regularly scheduled web-site. All "tl;dr" will be stoically acknowledged.