Genèse 12:10

And there was a famine in the land, and Abram went down into Mizraim to be a dweller there, because the famine was strong in the land.

AND THERE WAS A FAMINE IN THE LAND. Now Abraham went down to Egypt on account of the famine to dwell there in order to keep himself alive in the days of the drought, but the Egyptians oppressed him for no reason [and attempted] to take his wife. The Holy One, blessed be He, avenged their cause with great plagues, and brought him forth from there with cattle, with silver, and with gold, and Pharaoh even commanded his men to escort them from the land. He thereby alluded to Abraham that his children would go down to Egypt on account of the famine to dwell there in the land, and the Egyptians would do them evil and take the women from them, just as Pharaoh said, And every daughter ye shall save alive, but the Holy One, blessed be He, would avenge their cause with great plagues until He would bring them forth with silver and gold, sheep and oxen, very rich in cattle, with the Egyptians pressuring to send them out of the land., 12:33. Nothing was lacking in all the events that happened to the patriarch that would not occur to the children.
The Rabbis have explained this subject in Bereshith Rabbah: “Rabbi Pinchas said in the name of Rabbi Oshaya that the Holy One, blessed be He, said to Abraham, ‘Go forth and tread out a path for your children!’ Thus you find that whatever is written concerning Abraham is also written concerning his children. In connection with Abraham it is written, And there was a famine in the land; in connection with Israel, it is written For these two years hath the famine been in the land.”
Know that Abraham our father unintentionally committed a great sin by bringing his righteous wife to a stumbling-block of sin on account of his fear for his life. He should have trusted that G-d would save him and his wife and all his belongings for G-d surely has the power to help and to save. His leaving the Land, concerning which he had been commanded from the beginning, on account of the famine, was also a sin he committed, for in famine G-d would redeem him from death. It was because of this deed that the exile in the land of Egypt at the hand of Pharaoh was decreed for his children. In the place of justice, there is wickedness and sin.

לגור שם, not to settle there permanently.

ויהי רעב בארץ, this was one of the ten trials Avram was subjected to by G’d. He successfully mastered all of them. He never second-guessed G’d, thinking that if G’d had done things differently it would have been better, especially, better for him. He could have said that G’d had said to him that all the nations of the earth would be blessed through his presence, and if so how did his presence help them if the now had to face a famine? He did not question

מצרימה, generally speaking, there was always plenty of food in Egypt even if famine was prevalent in neighbouring countries because Egypt did not depend on rain from above, but relied on the waters of the river Nile which were most dependable both in quality and in quantity. Even the Torah mentions this as an advantage of Egypt in Deut. 11,10 when comparing the land of Israel to that of Egypt. Avram could not cope with the famine in the land of Canaan, due to all his herds and flocks requiring water and grazing land. Besides, by then he had a household comprising many people for whose welfare he was responsible.

לגור שם, he did not intend to settle in Egypt, only to wait out the famine there.