Bells theorem says that he predictions of quantum mechanics are incompatible with any "local hidden variable model". Experiments show that the real world appears to follow the predictions of quantum mechanics, so no local hidden variable model can explain what we see in these experiments.
Therefore all you need to understand is what a "local hidden variable model" is, and what the lack of one would mean.
You should think of doing measurements in quantum mechanics as something like flipping a biased coin, or rolling a biased dice. Every time you do a measurement you get some outcome (maybe heads or tails or a number from 1 to 6) but the probabilities aren't generally the same, they're biased in a way which tells you something about the system. They can even be biased so much that the outcomes are deterministic (0 and 1 are still probabilities).
When you flip a coin or roll a dice, the outcomes are random from your point of view, but there isn't really any fundamental randomness going on. There's a bunch of data about how your hand moves, and the local wind speed and pressure and the physical characteristics of the coin which if you knew them then you'd be able to predict the outcome with certainty. People have even been able to build robots which can reliably flip coins to always get a desired outcome.
This "extra information" that makes the outcome deterministic if you know it we could call hidden variables. Its "hidden" because you don't generally know it.
If Bell's theorem said that there was no hidden variables model consistent with the predictions of quantum mechanics we'd be done now. We would have decided that quantum mechanics is fundamentally probabilistic and no extra hidden information could exist to make it deterministic. But Bell emphatically doesn't do that. The theorem says there is no local hidden variable model consistent with quantum mechanics. So what does "local" mean here? For that you can imagine that instead of rolling one dice I'm rolling one here, you're rolling a second one wherever you live, our friend on Mars is rolling a third one, etc. A hidden variable is "local" if it only affects the dice in once location.
Bell's theorem says that if you want to have a hidden variable model that reproduces quantum mechanical predictions then it needs to be nom-local. That is the hidden variable has to affect the experiment here, and the one with you, and the one on Mars all at the same time. We consider that non-local hidden variables are unlikely because they're essentially magic, they need to affect things arbitrarily far away from each other all at the same time. Amongst other things this is wildly on conflict with special relativity.
Unlikely does not mean the same as impossible, so no.
There are also some "loopholes" in the story I told above. For example the many worlds interpretation of quantum mechanics is completely local and deterministic, but weird enough that the standard formulation of hidden variables models doesn't apply to it.
