The judiciary has the power to interpret the United States Constitution because it took that power for itself in the case of Marbury v. Madison, decided in 1803.
The Constitution itself does not make clear whether the judicial branch has the power of judicial review (the power to determine whether acts of Congress are constitutional). The Constitution simply states
The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution…
This could be taken to mean that the judiciary has the power of judicial review, but it does not necessarily have that meaning.
In 1803, the Supreme Court decided Marbury. In that case, the Court ruled that the Judiciary Act of 1789, under which Marbury was claiming that he had the right to a writ of mandamus, was unconstitutional. In his decision, Chief Justice Marshall wrote that
if a law be in opposition to the Constitution, if both the law and the Constitution apply to a particular case, so that the Court must either decide that case conformably to the law, disregarding the Constitution, or conformably to the Constitution, disregarding the law, the Court must determine which of these conflicting rules governs the case.
By saying this, he implies that the judicial branch can say that a law is in opposition to the Constitution and that it must be disregarded. In order to do this, the judicial branch must have the right to say what the Constitution means. Thus, the judiciary has the right to interpret the Constitution.
The judicial branch, then, took the power of judicial review for itself in Marbury.