What is the most likely/favoured theory put forth explaining the Black hole information paradox? Is there any theory that is most favoured that has no flaws, or most likely explains how black holes don't destroy information?
There is no favored one, still a controversy. The two comments before this was posted talk about two ways.
Another, and one eliciting a lot of research, is the holographic principle, or AdS/CFT which says that the information or physics in n dimensions is encoded, i.e. fully there, in the n-1 dimensional space (or spacetime) that is its boundary. So the information of a BH, and of course anything that arrives there, would then be in its horizon. It could be non-contradictory with the firewall explanation, just some physics as to how. Google it.
Another is that there is in fact, information that comes out in some version of the Hawkings radiation. This one is more complex to explain or debate, but it says that there is some non-random radiation, that is in fact entagled with the earlier information that was thought destroyed. This could also be not inconsistent with the holographic principle. There's been calculation and analysis done that said earlier that this was impossible,but some people are finding ways out
In reality the thought is that we won't know until there is an established quantum gravity theory, because even at the horizon there should be quantum effects that could be non-locally related correlated to the very strongm gravity near the singularity (remember the horizon could be huge, and have much weaker gravity).
What nobody seems to want to believe is that information can be lost. Physics has shown that if that can happen there is no causal structure to spacetime, that anything can then appear somewhere without any cause, and more or less equivalently there would be closed timelike curves (you could be the cause of your own birth).
The belief is less strong on the equivalence principle not being valid; if invalid there would be reason to think that strange things can happen at the horizon.