What are the implications of "Lema'an Yirbu Yemeichem ... "?
Rashi #1: It implies that if you keep the MItzvah of Mezuzah you will live a long time in Eretz Yisrael; otherwise, you will not. 1
Rashi #2 (in Shabbos, 33b): With reference to Pasuk 19 ("Velimadtem osam ... ") it implies that if you don't learn Torah and teach it to your children, you will not live long in the land.
Ramban #1: It comes to teach us that, although one is obligated to observe the Mitzvos even in Chutz la'Aretz when we are in Galus, the main Mitzvah is to observe them in Eretz Yisrael.
Ramban #2: It is coming to inform us that we will return from Galus and remain forever in Eretz Yisrael.
Brachos 8a: It says, "Yirbu Yemeichem
Rashi: Because the Torah is expounded by implication - positive from negative and negative from positive.
Kol Eliyahu: Why was R. Yochanan satisfied after he heard? The verse implies that long life is only in Eretz Yisrael! In the future, Batei Kenesiyos of Bavel will be fixed in Eretz Yisrael (Megilah 29a), therefore it is as if they are in Eretz Yisrael.
What is the meaning of "ki'Yemei ha'Shamayim al ha'Aretz"?
Ramban: It either means for all generations or it refers to 'the original heaven and earth at the time of the creation and the days of the world'. 1
Rashbam: It means that the rain will descend from the Heaven and the earth will produce its fruit.
Lev Eliyahu (Shemos p.92): Through the Mitzvos, you will have great days like Shamayim, i.e. purely spiritual, while on the ground.