Why does Don Corleone give in so easily in The Godfather?

I think this is made clearer in the book than in the movie. Vito Corleone is a very patient and calculating man. He is willing to wait and slowly manipulate things, unlike his angry and rash son Santino.

At the time of Santino's (Sonny) death Vito realizes that an all out war is costing him too much (the death of his son). He decides to stop the killing by meeting with all the families and agreeing to the drug trafficing. He also learns that the Barzini house is ultimately behind a plot to destroy his family.

In the book (where you can see into the minds of the characters) it is obvious that Vito Corleone doesn't forget or forgive insults. It may take him years, but he eventually enacts his revenge. He likely saw his time was over, given his failing health, and so decided to invest what time he had left into training Michael to take over (and ultimately get revenge). Santino was the oldest and the natural successor, even though he didn't have the temperment to be a good leader. Fredo was weak willed and less intelligent, particularly after the attack on his father (before that attack he was seen to be very tough). Michael was the only son that had the calculating mind of his father and the patience to wait for opportune moments.

In book and the movie Michael Corleone kills the heads of the houses and their co-conspirators (such as Moe Greene). The book concludes with the Corleone family once again ascerting themselves as the most powerful family in New York. The movies take a different turn, with Michael attempting to legitimize the business and get away from crime. Neither the book or the movie is clear about whether drug traffficing continued after Michael destroyed all his enemies. It is possible he reversed that position once he was the only one left standing.