What is the significance of the phrase "Vay'hi ki Heicheil ha'Adam Larov al-P'nei ha'Adamah"?
Ramban: It serves as an introduction to the chapter of the Flood, 1 and the Torah is telling us here that the people already began sinning early on, when Noach was four hundred and eighty. That was when Hashem decreed that they would all perish in the Flood, and He was merely waiting for their measure to be full before destroying them. 2
Peninim mi'Shulchan ha'G'ra: It connects to the end of the Pasuk, "u'Banos Yuldu Lahem". Girls bring increase to the world, since only they give birth. A boy brings no increase. Even if he marries and has children, his wife could have married men who were already born! 3
Bava Basra 16b: Two Amora'im discuss the meaning of the word "la'Rov" - one explains it literally to mean that the world's population increased; the other, as an expression (from 'la'Riv' - that quarreling came into the world. 4
Oznayim la'Torah: It teaches us that the decline of man was commensurate with the increase of mankind, because as long as they were few in number, the people felt weak and vulnerable; but as they began to increase, they became more confident and arrogant ('strength in numbers').
Oznayim la'Torah: Like we find in Melachim II 17 prior to the Galus of the ten tribes, and in Divrei Hayamim II 36 prior to the Galus of Yerushalayim.
That is how Rebbi consoled his son, when a daughter was born to him. Bar Kaparah said that it is vain consolation (Bava Basra 16b), since it is good for the world, but bad for the father. A daughter can have a limited number of children. A son can marry many women, and father so many children!
See Torah Temimah, note 1, who explains the Machlokes.
