The Supreme Court's interpretation of this right in this case was that the right of habeas corpus applies even to people who are not citizens of the United States so long as they are being held in a place that is completely under the control of the US.
Writing for a bare, 5-4 majority, Justice Anthony Kennedy emphasized the importance of the fact that the US has de facto control over Guantanamo Bay. Because of this, people who are at that base are covered by the Constitution. Kennedy also points out that the right of habeas corpus is one of the very few that is enshrined in the pre-Bill of Rights Constitution, thus showing that it was a very important right to the Framers of the Constitution.
The Court, then, held that the right to habeas corpus is a very important right that applies to all those under US control unless a compelling reason is given to suspend that right.