With regard to the brief of a case, the analysis is a short discussion of how the court came to the ruling that it made in a given case. You are essentially meant to explain the court’s ruling. In this case, the analysis is basically that the Supreme Court felt that there could be no meaningful protection for people under the 4th Amendment if there were no exclusionary rule.
The 4th Amendment protects us from unreasonable searches and seizures. Police are not allowed to search or seize our property without probable cause to believe that they can find evidence of a crime if they conduct a search. However, what does this protection really mean if evidence that is uncovered through an illegal search can be used against us in court? If such evidence can be used, there is little incentive for the police to obey the 4th Amendment. Therefore, if the exclusionary rule is not applied to the states, the 4th Amendment will essentially not protect us from anything.
The Court’s rationale, then, was that the exclusionary rule was necessary in order to make the 4th Amendment have any real meaning for us.