Why do the justices disagree on the meaning of the Constitution?

The justices disagree based on their own personal and political believes. Justices are appointed by presidents, which results in justices coming from different backgrounds and political believes. The diferences in political part, backgrounds and belifs reutls in differing opinions when it comes to the meaning of the constitution.

Explanation:

The  Supreme Court  takes its powers from Article III  of the Constitution.  Article III, §1  provides  that  "the  judicial power of the United States,  shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish."  In accordance with this provision, the Suprem Court of the United States was created by the authority of the Judiciary Act of 1789.  The Court met for the first time on February 2, 1790.

The Court currently consists of a chief justice and eight associate justices, each with equal voting power to the chief's.  (The number of justices has varied, beginning with six, then increasing to seven in 1801, and finally to nine in 1869.) Each justice is nominated by the President, confirmed by the Senate, and serves for life.  The Senate confirmation process begins with hearings before the Judiciary Committee and ends with a vote of the full Senate.  A simple majority is required for confirmation.  Justices who commit "high crimes or misdemeanors" are subject to impeachment and removal from office.  

The Term of the Court begins, by law, on the first Monday in October.  In a typical year, decisions are announced in all the argued cases by the end of June.  An average of about 7,000 to 8,000 petitions are filed with the Court over the course of a single Term; of those only about 100 or so are set for full briefing and oral argument.  In addition to the petitions, another 1,200 or so applications (e.g., a request for an immediate stay of a decision below) are filed each year.  These applications can be acted upon by a single justice.