Why does Voldemort kill Harry's parents?

In 1980, when Sybill Trelawney was having an interview with Albus Dumbledore in the Hog's Head pub for the position of Divination Teacher, Dumbledore at first thought she was a fraud. Then, suddenly, she went into a trance and made a prophecy that a child would soon be born who would defeat the Dark Lord. She said that the child would be a boy born at the end of July and that he would have powers that could pose a challenge to Voldemort. Severus Snape, who was in the pub at the time, related this prophecy to Voldemort, who determined that the child in question was Harry Potter, particularly because of the element in the prophecy that said the child's parents would be people who had previously defied Voldemort. Ultimately, of course, we learn that the prophecy could equally have applied to Neville Longbottom, but the Potters were better known to Voldemort—and Harry, like Voldemort, was a half-blood.
When Dumbledore found out that Voldemort had such designs, he suggested the Potters go into hiding and use a secret keeper. A complicated plot was enacted wherein the Potters pretended that their friend Sirius was the secret keeper, but in fact they entrusted Peter Pettigrew with the secret, not knowing that he was working for Voldemort. Ultimately, Pettigrew betrayed his friends to the Dark Lord.
When he found out the Potters' whereabouts, Voldemort set out to kill the baby before the child could defeat him. When he arrived at the Potters's home, James told Lily to take the baby and run, but he could not protect them—Voldemort killed him immediately with a curse in order to get to Harry. He did not actually intend to kill Lily. In fact, he told Lily to step aside, and said "this is my last warning," indicating that if she did not do so, he would kill her. In the event, Lily does not step aside, and her act of sacrifice for her son protected him.

The short answer to your question is that that Voldemort killed Harry's parents because they were in the way. When Sybil Trelawney was interviewing for her eventual job at Hogwarts, she made a prophecy to Albus Dumbledore that "the one with the power to vanquish the Dark Lord approaches." The specifics of the prophecy describe the general time when this person would be born and offer some information about the person's parents. At the time of the prophecy, Severus Snape was eavesdropping and heard the prediction. Although he was discovered by Dumbledore, he still went to the then-powerful Voldemort to reveal what he heard. The prophecy scared Voldemort, who then targeted Harry Potter as the child that the prophecy described. When Voldemort went to kill Harry, Harry's parents tried to protect themselves and their son. Harry's parents died in the process, but the love of Harry's mother, Lily, protected Harry and allowed him to survive the attack, thus paving the way for his future and the eventual defeat of Voldemort.