A pretty simple explanation that should be mostly fine:

Suppose you have a "magic" black box that produces entangled particles with spin up and spin down. One is always up and the other is always down.

If you measure one in up/down base, you will get up in 50% of cases and down in the other 50%, while the other particle will be always just opposite. You can easily argue that here, you merely didn't know what came out and when you measured one you just figured out what both were. Everything works just fine, no need for any QM yet.

But suppose you measure this setup in left/right base instead. Then, obviously, the first spin you measure will have 50% to be left, and 50% to be right, doesn't matter if it was up or down before. Now, predictions for the second spin are different. In classical physics, you are measuring left/right with 50% for the other particle too. But QM entanglement says that nope, we know with 100% certainty that the second spin will be right if the first was left and vice versa.

This can be tested, has been tested extensively in various configurations and QM entanglement was found to match reality, while the classical physics doesn't.

You STILL have several possibilities:

1. Typical QM interpretation is right that stuff doesn't have state until measured.
2. When you measured one particle, you changed both particles instantly.
3. Everything including the measurement outcomes had been predefined before you even thought to make the box to measure this.
4. Maybe something else.

But you can't have good old classical physics where stuff behaves as you would expect and want it to.

A pretty simple explanation that should be mostly fine:

Suppose you have a "magic" black box that produces entangled particles with spin up and spin down. One is always up and the other is always down.

If you measure one with up and down base states, you will get up in 50% of cases and down in the other 50%, while the other particle will be always just opposite. You can easily argue that here, you merely didn't know what came out and when you measured one you just figured out what both were. Everything works just fine, no need for any QM yet.

But suppose you measure this setup in left/right base instead. Then, obviously, the first spin you measure will have 50% to be left, and 50% to be right, doesn't matter if it was up or down before. Now, predictions for the second spin are different. In classical physics, you are measuring left/right with 50% for the other particle too. This happens because you have 50% it was up * 50% it gets measured right + 50% it was down * 50% it gets measured right (same for left) But QM entanglement says that nope, we know with 100% certainty that the second spin will be right if the first was left and vice versa.

To summarize: measure in up/down axis: both give 50% for each value of the first particle and 100%/0% for the second particle. Measure in left/right axis: both give 50% for each value of the first particle. Classical physics gives 50%/50% for the second one, while QM gives 100%/0%.

This can be tested, has been tested extensively in various configurations and QM entanglement was found to match reality, while the classical physics doesn't.

You STILL have several possibilities:

1. Typical QM interpretation is right that stuff doesn't have state until measured.
2. When you measured one particle, you changed both particles instantly.
3. Everything including the measurement outcomes had been predefined before you even thought to make the box to measure this.
4. Maybe something else.

But you can't have good old classical physics where stuff behaves as you would expect and want it to.