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

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.

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?