Can you give a complete physical description of why Lego blocks fit together in particular ways? What's the fundamental physical interaction(s), in detail, then make it so some Legos can fit with other Legos, and some Legos can't?

You can't. Actually, nobody can, because we don't have a coherent theory that is known correctly predict all of the interactions, at all of the scales, which are involved in two Legos sticking together. However, that doesn't prevent you from experimenting with Legos and observing that Legos come in a variety of sizes and shapes, and some of them can stick to other Legos in one particular way and some of them can stick in different ways. This is how people discovered things through experimental chemistry: they had atomic theory, which helped provide insight at an important level into the structure of everyday substances, but they didn't need quantum chemistry to experiment with bonding and breaking bonds and draw logical conclusions from experimental results.