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

For Don Corleone, there were certain red lines you don't cross, and poisoning the people of your neighborhood with drugs was one of those red lines. It also didn't help that Sonny spoke up at the meeting the way he did. Essentially, Don Corleone realized that he had no leverage left among the other dons - they knew not only what his position was, but also who they could potentially exploit to get around that.

Spoiler ahead for further explanation:

Had Carlo not had Sonny killed in the tollbooth, it's entirely possible Sonny would have been convinced to go along with dealing drugs, even after the attempt on Don Corleone's life. Michael was the one wild card none of the other families knew what to make of him; as far as they were concerned, he was either weak like Fredo or just a simple civilian. That was what allowed them to think they could fool him into sitting down with a meeting with Barzini at the end of the film - they just didn't know that Don Corleone had warned him about that possibility before he died.

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.

Don Corleone's first priority was his family's safety - especially Michael's. By signing peace treaty with others he ensured no one harmed Michael in any way and interfere with his homecoming.

But he was never weak as at the same time he warned others that he would not leave anyone who would try to harm his family -

"But I'm a superstitious man. And if some unlucky accident should befall him - If he should get shot in the head by a police officer, or if he should hang himself in his jail cell - or if he's struck by a bolt of lightning, them I'm going to blame some of the people in this room, and that I do not forgive. But, that aside, let me say that I swear, on the souls of my grandchildren, that I will not be the one to break the peace we've made here today."

Also, he knew that he alone cannot go against Tattagallia and Barzini as they may seek support of other Dons. And none of the Dons at that time wanted an all out open war, so they may choose the stronger side which was apparently not of Don Corleone at that time.

Vito had his own agenda. Allowing other families to fight over the drug revenues would occupy their resources and weaken their ranks. He got the assurances he sought. He bought Michael the time and opportunity to go after Las Vegas gambling, rackets, prostitution etc. This also afforded the time to see how the other families reacted to the drug free-for-all. In the end, the other 4 families lost sight of the big picture. Michael took out the heads of the other four and Moe Green. Checkmate!

Vito had no choice. Barzini had a lot of influence in certain parts of Sicily and he was beginning to sniff Michael out (as evident when Fabrizzio planted the bomb that killed Michael's wife). Also he promised to keep the peace, and by doing that he bought himself some time to train Michael to head the family business and connect him to all the important political figures (which was where half of the Corleone strength came from) so he can finish the war. That way Vito would not betray his honor by lying, since his word is not binding to Michael.

Don Corleone traded the rights to deal drugs (with controls) in New York for the right of Michael's safe return.

An important part of the scene is when Nazorine -- a kind of neutral Don -- speaks-up regarding the dealing of drugs. "My people can't refuse." This shows which way the wind is blowing -- towards drug dealing.

Don Coreleone can reluctantly live with his painful New York concession because he intends to move the family away from the "dirty business" of drugs in New York to the "innocent vices" of gambling and women in the casino business of Las Vegas. Of course, getting Michael back to the USA is paramount, thus the lengthy warning speech about Michael's safety.

The other Dons are very happy with this deal. Even after the Turk's death, they finally get to run their lucrative drug business in New York under Corleone's protection. They understandably don't perceive/recognize the return of Michael and his abilities as a threat. They just have to count down the days for Vito to die and then they won't even have to pay a concession to the Corleone's anymore.

Vito needed to make peace so Michael could return from Sicily safe. Otherwise he would be killed on one way or other. Even after the peace Brazinis tried to kill Michael but it was much safer with the peace...