The significance of this case was that it established the idea that a person has the right to sue government agents for violation of his or her Fourth Amendment rights.
In this case, Bivens sued the FBI agents for damages when they searched his house without probable cause. The lower courts dismissed his suit because there was no law that gave him the right to sue. When the case reached the Supreme Court, the Court ruled that the right to sue the agents was implied by the very existence of the 4th Amendment.
This is significant because it gives people a way (other than the exclusionary rule) to deter police from violating their rights.