Urea acts as a chaotrope when in the presence of folded proteins and enzymes. It's not something you want floating about willy-nilly in your body. The structure of the molecule allows it to sow utter chaos:

:O:
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C
/ \
../    \..
H2N      NH2

Those two pairs of electrons hanging out at the tail make the oxygen end of urea extremely electronegative and eager to bond. As it turns out, the hydrogen bonds that stabilize the outer shell of a folded protein are the perfect places to steal atoms from to solve urea's vexing problem. The chaotrope occupies hydrogen bonds, leaving the outer structure of the protein somewhat unstable. Hydrogen bonds are certainly not the only forces that hold together quaternary structure, however. The hydrophobic effect is also important, the desire of nonpolar portions of a protein to be as far away from surrounding water as possible, therefore rolling themselves into a little ball inside the protein.

Without the safety of a stable external structure to protect them, the hydrophobic portions of the molecule are soon assaulted with molecules of water worming their way inside. Once this happens, all bets are off. The protein increasingly loses every anchor point that allows it to maintain its distinctive shape; it denaturizes. What was once an extremely complex and useful association becomes nothing more than a chain of amino acids again.