The Biblical meaning of the colors of the rainbow are that red signifies sin, orange means evil, yellow stands for eternal life and purple signifies loyalty. In addition, in the Biblical context, the colors of the rainbow taken as a whole signify God's covenant with man to never again destroy the world with a flood.
In the Bible, the color red typically stands for sin and the blood of Christ that offers redemption. The Bible also discusses red as a color of suffering in the books of Exodus and Isaiah. By contrast, the color orange represents judgment upon human actions, as Biblical authors discuss in both the books of Psalms and Revelation. Pottery in the Bible is often orange, and Biblical characters occasionally smash pottery to express their disgust at human behavior.
The color yellow stands for eternal life and glory. In many points in the Bible, a yellow glow signifies Godliness and spiritual status. In addition, in the book of Job, the color yellow stands for bitterness.
Purple in the Bible stands for royalty. Kings in the Bible often dressed in the color. In addition, high priests of the Bible used the color purple for the Gate of the Tabernacle. In the Bible, purple often stands for God's justice and His plan to bring justice upon the world.
As a whole, God displays the colors of the rainbow at the end of the flood that destroyed the world. The colors combined stand for the power of God, according to the book of Ezekiel. When God shows the rainbow to Noah after the flood, it is His sign that God's plan is to never again destroy the world with a flood.