While what you say is true, it doesn't have anything to do with quantum mechanics. I'm sure the ancient Greeks would have had no problem understanding that if you put a red ball in one box and a blue ball in another box then if you open a box and see a blue ball you'd know that the red ball is in the other box.

In quantum mechanics, entanglement is more about the fact that until a measurement is performed on one of the balls, both balls are simultaneously blue and red and will behave as if it was in a superposition of both colors up until a measurement is performed. By extension, any other property that depends on the color of those balls will also be entangled with the balls and behave in a superposition of whatever properties are entangled.

For example if a washing machine washes clothes with hot water if the blue ball is in box A, and washed with cold water if the blue ball is in box B, then that washing machine will be washing clothes with hot and cold water until a measurement is performed on any of the balls or the washing machine itself. The clothes being washed will simultaneously be expanding (from hot water) and contracting (from cold water).

Only once a measurement is performed on any part of the entangled system will every property of the system collapse into a definite state of red or blue, hot or cold, expanded or contracted.

Thanks for the explanation!

How can we know anything about quantum entanglement when measuring the entangled particles, or any effects they have, cause the whole thing to collapse?

One straightforward way is you perform the same experiment over and over, the quantum weirdness will show up in the statistics. Eg you might test if two measurements are the same more often than they should be by the laws of classical probability.

(My understanding; I only took one course in quantum mechanics)

Arguably the information that you "gained" was predetermined when you decided which colored ball to place in which box. You didn't gain any knowledge that you couldn't have known before you were teleported 5,000 ly away.

What's also more difficult is that the balls wouldn't have definitive colors, but exist in a probabilistic state / superposition. But you couldn't send the message "I have a red ball" to the individual 5,000 ly away because collapsing your ball's wave will be random and you can't determine which color you want. You could "mistakenly" collapse your ball to blue, not red.

This big missing piece here is that it's not a simple red/blue property. Spin is the easiest to understand. When you measure, the spin will either measure up, or down. The key though is you can measure spin against any angle. If you measure the spin of the two particles at the same angle, you will get opposite answers. And you can choose what angle to measure at* long after the particles have separated.

* There is one theory of QM that removes the possibility of choice from the universe.

I talked about this elsewhere in this thread (just with gloves instead of colored balls).