Governor George Wallace is best known for the fact that he played on racism and opposition to racial integration to gain and keep political power. This is why he called the 14th Amendment "infamous and illegal."
Wallace called it "infamous" because it required that African Americans be treated as the equals of white people. This was against his beliefs and those of many Southern whites. He called it "illegal" because the Southern states were forced to ratify it before they would be allowed to have democratic government again after the Civil War.
By criticizing the 14th Amendment in this way, Wallace was expressing just how strongly he opposed civil rights and integration. He saw this as a good way to get people to vote for him.