People with insulin resistance or diabetes are more likely to have higher levels of insulin in their blood. That insulin can lead to increased production of fatty acids in the liver leading to more pockets of fat in liver cells (hepatocytes).
A fatty liver by itself doesn't damage liver cells. But it's sometimes associated with liver inflammation, which can be damaging. The Liver Foundation describes the causes and treatment of fatty liver disease.
How did soy protein protect against fat buildup in the rats' liver? It reduced the expression of certain genes responsible for fat production. It was also capable of increasing the breakdown of fat.
For a summary of the research:
Soy Protein Reduces Effects of Diabetes on Liver
For the research itself:
Appears in the Journal of Lipid Research, September 2005 (abstract only, subscription required for full article)
Soy Protein Reduces Hepatic Lipotoxicity in Hyperinsulinemic Obese Zucker fa/fa Rats