What is the significance of the phrase "And it was when the man began to increase in the land"?
Ramban: As an introduction to the chapter of the Flood, the Torah is telling us here that the people already began sinning early on, when No'ach 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 1 .
Peninim mi'Shulchan ha'Gra: It connects to the end of the verse "u'Vanos Yuldu Lahem." Girls bring increase to the world, for 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! 2
So Rebbi consoled his son, when a daughter was born to his son. Bar Kaparah said that it is vain consolation (Bava Basra 16b), for 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!