Because his father was a Muggle.
Yes, Lord Voldemort who sought out to destroy the Muggle realm is a half-blood.
His mother, a witch, was in love with the Muggle Tom Riddle and used a love potion to make him fall in love with her. They eloped and soon she became pregnant and thought that Tom truly loved her so she stopped giving him the potion, but he was horrified and fled back home, leaving her alone with a baby on the way. She gave birth to Tom Riddle II, Voldemort, and soon died of a broken heart. As Tom Riddle, Voldemort, grew up and was accepted into Hogwarts he learned of his parents and his fathers actions that lead to Voldemort being raised in the orphanage. Basically, he hated his father, who was a Muggle, so by extension, Voldemort hated all Muggles.
Not to mention that he was conceived under a love potion, so he never knew what love was and never experienced it.