TL;DR: Passages in the Bible say so.

Revelations 19:13 says that when Jesus returns, "He is dressed in a robe dipped in blood, and his name is the Word of God." The red sash symbolizes the blood that Christians believe Jesus shed on the cross to save humanity from sin. I'll edit to add the source if I find it, but red was also a difficult and expensive color to produce, so was associated with royalty.

The white robe comes from Mark 9:2-9, the scene in the Gospel of Mark that talks about the transfiguration of Jesus before the apostles. This is a significant event in Christian tradition because it is said that Moses (as a personification of God's Law) and Elijah (representing the prophets) stood with Jesus, who became radiant with light. The voice of God then proclaimed Jesus as the Son of the Father and commanded the apostles to listen to him. According to Christians, this is where heaven met humanity, with Jesus at the intersection. White has traditionally been associated with light and purity, which is why Christian brides have worn white (especially since Queen Victoria made it popular).

Here's the passage from Mark (full passage here):

2 After six days Jesus took Peter, James and John with him and led them up a high mountain, where they were all alone. There he was transfigured before them. 3 His clothes became dazzling white, whiter than anyone in the world could bleach them.

Red and white together symbolize the purity of Jesus the Christ, who came to cleanse the world from sin; the light of God shown to followers of Christ; and the blood spilled to give humanity eternal salvation.