Long story vastly over-simplified, it's thought that melamine is the only direct ingredient causing this entire problem. The secondary chemical involved (cyanuric acid) can be created from melamine by certain (otherwise harmless) bacteria as a byproduct. Recombining the acid and melamine under certain conditions (common in many kidneys around the world, and in some cooking methods also common around the world) causes crystalline build-up in the kidneys.
Kidneys can't repair themselves over time.
Melamine causes a material to read dramatically higher in protein than it actually is when subjected to commonly-done 'cheap and quick' tests. It also appears to be harmless by itself in higher-order organism metabolisms aside from increased urination.
High-protein often equates to high-value. And the buyers only do the 'dump a scoop in, press a button, wait five minutes, look at digital readout' tests the melamine is practically targeted at.
The real news-worthy bit nobody seems to be picking up on loudly enough: Much of the 'tainted' material involved in these pet-food recalls was being marked human-grade. Meaning it could have been used in your pizza crust. Or your burger bun. Go read blueroo's well-documented bit for more, and focus comments here.Link