It doesn't make sense unless you expand the quote a little:
But, in a larger sense, we can not dedicate -- we can not consecrate -- we can not hallow -- this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced.
Lincoln explained that there was no point for those who were alive (i.e. hadn't died in the battle) to even try dedicating, consecrating, etc., since the world wouldn't wouldn't care much what was said. Instead ("rather"), they ("the living") needed to focus their efforts on making sure that the soldiers killed in that battle hadn't died in vain; that instead of saying things, they (the living) should be doing things, i.e. preserving freedom and government of the people, by the people, of the people.
"For us" in this context means, roughly, "up to us."
As in "It's for you to decide where you want to study English."