The decision in this case has to do with what federal courts must do in cases where they are petitioned for a writ of habeas corpus. The courts are told in this case that they must conduct full hearings to determine the facts of a case if those facts were not determined by the state courts.
In this case, the Supreme Court held that the federal court needed to hold a full hearing to address allegations of fact by the man who petitioned for the writ of habeas corpus. The situation was that the state courts had not made any decision regarding the validity of this factual claim. In such circumstances, the Court said, the federal courts had the duty to hold a hearing to determine facts. This is generally not the duty of an appellate court.
The significance of the case, then, is that it expanded the rights of prisoners to appeal their convictions.