What does “the source does not by itself establish the broader causal proposition” mean?

This is an important methodological distinction.

Suppose the Library of Congress provides evidence that:

formerly enslaved people faced hostile whites after emancipation.

That establishes evidence of post-emancipation hostility.

But if your broader sentence is:

“American society had been structured for generations around racial hierarchy, and therefore abolition did not immediately transform social attitudes toward Black people.”

That contains a causal relationship:

historical racial hierarchy and persistence of social attitudes after emancipation.

A source documenting hostility after emancipation does not necessarily establish the entire causal chain.

It gives you evidence consistent with the argument.

To establish the broader causal proposition, you ideally want a historian who explicitly analyzes the relationship between:

slavery and racial hierarchy and post-emancipation social attitudes.