What did Abraham Lincoln mean when he said, "wringing their bread from the sweat of other men's faces ... "?

He would sometimes use a translated version of the bible, in this case, he did. This speech is an allusion to his bible, and he means that slave owners get labor produced yet it is not from their own work. Thus it was from the sweat of other men that that they obtained there "bread". There bread symbolizes all of their wealth.

Each looked for an easier triumph, and a result less fundamental and astounding. Both read the same Bible and pray to the same God, and each invokes His aid against the other. It may seem strange that any men should dare to ask a just God's assistance in wringing their bread from the sweat of other men's faces, but let us judge not, that we be not judged.