How does the supreme court affect rights?

They interpret the Constitution and any laws they may be deciding on, and determine if those laws violate anything that is in the Constitution. Many of those cases deal with the rights the Constitution grant to the citizens of the U.S. For example, a recent decision that dealt with the 2nd Amendment (the right to bear arms) said that the law in Washington DC that banned most people from owning handguns violated the 2nd Amendment right to bear arms.

Also, the Constitution does not only outline the rights of the citizens, it also outlines the powers of the different branches of government, and the Supreme Court also rules on this. An example of this is the Line Item Veto that Clinton had as President. It was challenged, and the Supreme Court rules it was unconstitutional because the power of the President, as outlined in the Constitution, was to veto an entire bill or pass it, not just veto a section of it and pass the remaining parts.