Why are we locked into 3-D space if universe if 4-(or-more)-D?

[Senegoid]

I’ve long wondered about this. I’m sure it’s a very trendy thing to wonder about.

Now, spurred to action by this post in this thread, I’m asking here.

@Chronos

We really do live in and experience a world with three spatial dimensions. All physicists agree on that. The holographic cosmology models, however, posit that the three-dimensional Universe that we experience is a holographic projection from a two-dimensional universe which is more fundamental. Meanwhile, there are also other models that posit that our Universe is actually just one slice of a higher-dimensional universe of some sort, with some sort of mechanism (there are a variety posited) preventing us from perceiving the extra dimensions.

If there are more dimensions “out there”, of which our known Universe is just a slice, what keeps us confined to our slice? What are some of the posited mechanisms? (More-or-less Dummy-level explanations, please.) I know of suggestions that there may be some extra “compact” dimensions laying around. But aside from that, are there models of the Universe with additional “large” dimensions?