Lee was coming off an absolutely devastating victory over the Union army under Hooker at the Battle of Chancellorsville. His decision to invade the north was based on a few things:

• He avidly read antiwar ("Copperhead") newspapers from the North, which led him to believe that the Union population was war-weary and ready to force Lincoln from office in the 1864 election and elect a president that would sue for peace.
• His army kept winning, but was chronically short of supplies, and he hoped to capture a bunch.
• Lee's later nemesis Grant was besieging Vicksburg, an important fortress on the Mississippi, and Lee hoped that if he raised enough havoc in the North's heartlands, Grant's army would have to lift the siege and send troops to stop him.
• Most of the war in the East up to that point had been fought in Virginia, which meant that Virginia homes were burned and Virginia food was eaten and so on, and fighting it elsewhere for a while was attractive to Lee.

Lee didn't know exactly where the Union army was because his cavalry had gone off raiding. Lee had no particular desire to fight at Gettysburg. His subordinate, division commander Harry Heth, claimed in his memoirs that he sent troops into the town because he had heard there was a shoe factory there, and many of his troops badly needed shoes, but some historians dispute this. But whatever the reason, the troops headed toward the town. Before they quite got there, they bumped into some Union cavalry and got tangled up fighting them. From that point, both sides (with poor coordination and without really understanding what was going on) started feeding more and more troops into the battle. Gradually, more and more senior commanders arrived at the battle and more and more troops got involved, until the thing kind of took on a life of its own. All through the battle, Lee and various other Confederate commanders considered breaking it off at various points and continuing their strategic maneuvering elsewhere. In the end, Lee decided to fight it out, and the rest you know.

Sources: On Lee's thinking:
Coddington, Edwin B. The Gettysburg Campaign; a study in command. New York: Scribner's, 1968. ISBN 0-684-84569-5.
See James M. McPherson, "To Conquer a Peace? Lee's Goals in the Gettysburg Campaign." Civil War Times (2007) 46(2): 26-33.

For the controversy surrounding Heth and the Shoes, see:
Eicher, David J. The Longest Night: A Military History of the Civil War. New York: Simon & Schuster, 2001. ISBN 0-684-84944-5