General Sherman and the war secretary met with leaders of the black community, and Sherman then issued a field order to promise former African-American slaves the 40 acres. Promise of a mule came later but both have come to be grouped together. The promise was later withdrawn during Reconstruction. (It's also the name of Spike Lee's film company.)
That was a new one on me too. I assumed it was an American one, looks like I was right. Though judging by how few got it, it appears to quite an unusual American one.
Never heard it, either, unless it refers to holding a carrot hanging from a stick in front of mule to get him to move, but that's carrot and stick, not "a" carrot and "a" stick.
It refers to ways of getting someone to do what you want (like getting a stubborn donkey to move) with some combination of positive incentives (luring with carrot) and negative incentives (a smack on the rump with a stick).