There's a few different ideas that it would help to separate.
1. EPR's original argument and Bell's inequalities
EPR's original argument was that quantum mechanics would make an absurd prediction about a hypothetical scenario involving entanglement.
Bell essentially crystalized this weirdness into a precise statement that can be tested by an experiment. Classical physics (more precisely, a theory of local hidden variables) would necessarily make a prediction that did not agree with the quantum mechanical prediction.
The experiment has been done, and the result conclusively demonstrates that quantum mechanics gives the correct result, and a theory of local hidden variables cannot.
This experimental result does not depend on your interpretation of quantum mechanics.
2. How to understand the EPR paradox in different interpretations of quantum mechanics
You can try to understand the results of measurements confirming Bell's inequalities in different interpretations of quantum mechanics. Each interpretation predicts the same outcome of the experimental result, so this exercise is purely philosophical, and does not give any solid, empirical reason to prefer one interpretation over another.
To be concrete, let's imagine we send two entangled directions in opposite directions. Alice measures the spin of one electron along some axis, and Bob measures the spin of the other electron along some axis.
In the Copenhagen interpretation, once Alice performs her measurement, the entangled state collapses. Bob's measurement is then determined by this new state. This does not violate causality, because Alice and Bob cannot use wavefunction collapse to send any information to each other superluminally.
In the many worlds interpretation, the wavefunction splits into a world where Alice measures one outcome, and another world where Alice measures the other outcome. In each world, Bob's measurement is determined, but again information cannot be passed superluminally from Alice to Bob or vice versa.
At the moment, there is no conclusive argument that compels you to choose one interpretation over the other. Personally, I think it is useful to be able to use the language of multiple interpretations, because sometimes different interpretations allow you to have different intuition or insight into a problem, but not to take any one of them too seriously. Carroll would likely argue that the philosophical properties of many worlds are more pleasing than other interpretations, and so we should accept its point of view over the Copenhagen interpretation.
3. The research program ER=EPR
ER=EPR is a slogan for a research program started by Juan Maldacena and Leonard Susskind in the following paper: https://arxiv.org/abs/1306.0533. It is a speculative idea to resolve the information loss paradox and other issues in quantum gravity by relating quantum entanglement with wormhole solutions in classical General Relativity. The idea is about the underlying math of quantum mechanics and GR, and does not depend on your interpretation of the math of quantum mechanics. This is simply because they are working within the framework of quantum theory, and quantum theory itself does not tell you how to interpret it.
