“Interpretations emphasizing other consequences or interests”

This means interpretations that focus on aspects different from the dominant narrative.

For example, instead of:

Lincoln is the liberator who ended slavery.

another interpretation might emphasize:

abolition also transformed the labor system;

or:

emancipation weakened the Confederate economic and military system;

or:

emancipation had consequences for capitalist labor relations;

or:

Lincoln's policies involved both moral and strategic considerations.

These interpretations do not necessarily deny the first one. They may add another dimension.