The meaning of the two suns above Earth is explained by the message on the screen, sent by Hal 9000, and by Heywood Floyd's ending soliloquy.
All these worlds are yours except Europa
Attempt no landing there
Use them together
Use them in peace
Narrator Heywood Floyd:
... the President of the United States ..., and the Premier of the Soviet Union ... saw the new distant sun in the sky. They read the message, and perhaps learned something, ... they ... recalled their ships and ... planes. ... I still don't know what the monolith is. Maybe an embassy for an intelligence beyond ours ... [Earth] will never know a sky without [the suns] ... We are only tenants of this world. We've been given a new lease ....
What these mean is this. The "embassy" power the Monolith represents has given worlds to humans and kept Jupiter's moon, Europa, to itself. The command is for Americans and Russians (symbolizing the whole world) to use and share the worlds in peace with each other. In a foreshadowed intent to honor this command, the President and Premier "recalled" their weapons of war. In other words, the "embassy" power orders peace as a condition for Earth's continuance, for its "new lease." Floyd indicates that Earth has had an epiphany (spiritual realization and awakening). Earthlings now know that we are to tend, cultivate, and protect Earth, not exploit and ruin it, as a condition of our survival, which can be terminated as there is a higher power, the "embassy of a higher intelligence," the "something without shape."