Perhaps it would be more accurate to say it was promised to Abraham by a god, since according to Priestly tradition in Biblical scholarship Yahweh, who Biblically is truly God, is first known by Moses 500 years after Abraham. Abraham then could not have known Yahweh. However, in the Yahwist tradition of Biblical scholarship Yahweh is known before His revelation to Moses at the "burning bush." But, He is known as one of many gods. Abraham, whether in the Yahwist tradition or the Priestly tradition, was not a believer in monotheism. He was a believer in monolatry, a worshiper of one of many gods.
Terah, Abraham's father, took him and his family out of their native land, Ur, modern day Iraq, for unknown reasons and moved them to Harran, modern day Turkey. The Biblical story says that God promised Abraham Canaan when he was in Harran.
The promise was passed on by word of mouth to Abraham's son Isaac. Isaac had two sons, Esau and Jacob, and the promise was passed on to Jacob. It was in Jacob's time that a severe famine hit the region and forced them to move to Egypt. They didn't leave Egypt for over 400 years. Then they wandered in the wilderness for another 40 years.
The conquest of Canaan under Joshua was the first time that Abraham's descendants started to take the land for themselves by violence.
The Priestly tradition of the Genesis story says that by the promise of some god, not Yahweh, the land belonged to Abraham before the Jews went to Egypt on account of a famine for 400 hundred years. But Abraham did not exert any violent claim on the land itself. In fact Abraham's way of participating in the promise and bringing it about was to buy with the consent of the owner and pay full value for the cave of Machpelah in Hebron as a family tomb. He made absolutely certain he purchased it for the full amount with which the owner would be content. Abraham didn't fight any of the Canaanites or Philistines for property rights in Canaan.