a 2-D outer surface gets mapped to the 3-D interior... assuming it even happens!
It's clearly not proposing that the 3-D interior of our universe would be like a photographic hologram that isn't tangible.
It appears to be a mathematical construct that was proposed after examining principles related to black holes and recognizing it could be applied to the everything else in the universe as well. The holographic model wasn't pulled out of thin air, but was derived from efforts to avoid contradictions of basic principles of physics in regard to black holes.
How it works, assuming it's an accurate model for everything, isn't well understood. The holographic model makes some different predictions, and that's what is being checked at this point.
For example, there's subtle differences in predictions related to something called cosmic inflation. That's something that supposedly happened in the very early universe that helps explain the almost uniform temperature of the cosmic microwave background radiation. It's a mathematical model that was created to explain the CMB observations. In the holographic model, cosmic inflation is a natural consequence but the effects should be slightly different in the CMB.
Something else that's interesting is that Einstein's General Relativity equations can be derived by first creating a holographic model based on quantum theory! Is it the long-sought merger between quantum theory and relativity? That's sometimes called quantum gravity.
Here's a talk by Tom Banks that delves into it without a bunch of math (skip to about the 45-minute mark for some of the derivations based on the model):
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