If you want one quote from Farewell to Manzanar that briefly answers this question, you can find it at the end of Chapter 4. In general, we can say that the shikata ga nai attitude affected the Japanese Americans’ response to the internment by allowing them to stoically accept the humiliation and injustice of their predicament. At the end of Chapter 4, the author tells us that, because of this attitude,
…they were able to take a desolate stretch of wasteland and gradually make it livable.
This attitude allowed them to take the bad situation they were in and deal with it.
On p. 16, we are told that shikata ga nai means “it cannot be helped” and “it must be done.” Literally, it means “there is no way of doing things.” “Shikata” means “a way of doing things” and “ga nai” means “there is none.” Figuratively, then, it means that there is no way to do things other than the way that exists. There is nothing that can be done to make things different.
This is clearly a very stoic attitude. When people are faced with unpleasant conditions, it is helpful for them to have such an attitude because it helps them deal with adversity. This is what this attitude did for the internees in Farewell to Manzanar.