I was reluctant about writing an answer for this question, with so many good answers already here, but there are some points that I would like to mention, and I think it is missing in this discussion.

First, Bell's Theorem is a no-go theorem. It means that it proves that a specific situation is physically impossible, using a proof by contradiction. The situation in question is the existence of Local Hidden Variables (LHV) explaining the correlations obtained in a Bell test that are produced by measuring over a shared quantum system in an entangled state. If you set the quantum state and the measurements properly, it is possible to show a mathematical contradiction with the assumption of the LHV and what is predicted by Born rule.

The theorem doesn't mean that all quantum correlations are non-local. We could find a lot of quantum correlations that are local. For example, if in the Bell test, you choose a product state or a separable state, it would be local, since entanglement is a necessary condition for non-locality (but not sufficient, since some entangled states are local either). However, since there are some correlations which is non-local, it rules out the possibility that someone could find one day a local model "reproducing quantum theory", or more precisely reproducing all possible quantum correlations in a Bell test.

Another important thing about Bell theorem is its theoretical meaning. I disagree with rob that the theorem just raise a question that was answered years later by the experimentalists. To get sense of it, think about this situation: Imagine that I decide to invent a new model for electromagnetism, alternatively of Maxwell equations and I decide to name it egocentrically Ruffolo equations. Now, imagine that Ruffolo equations have very nice features, but it predicts that a point charge would have a field that is no spherically symmetric, contradicting Coloumb's Law and what Maxwell equations predicts.

We can say that the right thing to do in this situation is to do the experiment. To test Ruffolo equations against Maxwell equations, one should try to build a point charge experimentally, which is impossible to do exactly. So we should charge a small object and try to observe the electric field of it as far as we could, so it could be approximately a point, and we should do it in many directions we could, to testify if it is at least approximately spherically symmetric or not. But we don't need to spend this money and time. Maxwell equations and Ruffolo equations are not in the same stage. The former are being experimentally tested for almost two centuries. We trust in Maxwell equations, so any new model should at least reproduce what it already predicts.

The Bell theorem is quite the same situation that I described above. The quantum postulates, whose nobody question about its precision and correctness, already conflicts with LHV. The difference between local models and the Ruffolo equations is that one could argue, at least in principle, that Bell made some additional assumptions that could not fit in an experimental setup. To address the concern about these additional assumptions, experimentalists started to build loophole-free Bell tests.

There are some claims about Bell's personal point of view and also the point of view of Adam Becker, who is the physicist in the video. In the very nice article "Hidden variables and the two theorems of John Bell" by David Mermin, we find his last words

To those for whom nonlocality is anathema, Bell's Theorem finally spells the death of the hidden-variables program. But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible. What Bell's Theorem did suggest to Bell was the need to reexamine our understanding of Lorentz invariance, as he argues in his delightful essay on how to teach special relativity and in Dennis Weaire's transcription of Bell's lecture on the Fitzgerald contraction. What is proved by impossibility proofs, " Bell declared, "is lack of imagination."

We don't need to take Bell's personal point of view as mandatory about his theorem. Curiously, Mermin left a last footnote in this article:

Although I gladly give John Bell the last word, I will take the last footnote to insist that he is unreasonably dismissive of the importance of his own impossibility proofs. One could make a complementary criticism 'of much of contemporary theoretical physics: What is proved by possibility proofs is an excess of imagination. Either criticism undervalues the importance of defining limits to what speculative theories can or cannot be expected to accomplish.

If you want to have a perspective about how the community thinks about Bell inequality violations, take the poll "A Snapshot of Foundational Attitudes Toward Quantum Mechanics" made by Maximilian Schlosshauer, Johannes Kofler and Anton Zeilinger, in the conference “Quantum Physics and the Nature of Reality,” held in July 2011 at the International Academy Traunkirchen, Austria. The question "What is the message of the observed violations of Bell’s inequalities?" led to the following answers:

About this result, the authors says:

The Bell inequalities are a wonderful example of how we can have a rigorous theoretical result tested by numerous experiments, and yet disagree about the implications. The results of our poll clearly support this observation.

In conclusion, Veritasium video is a good introduction, but it is partial about advocating one perspective of the theorem. It is partially caused by Adam Becker personal point of view about it. In literature, we could find other perspectives about its meaning.