The Sages taught: Three sounds travel from the end of the world to its other end, and these are: The sound of the sphere of the sun, and the sound of the bustle of the crowds of Rome, and the sound of the soul at the moment that it leaves the body, which should be audible throughout the world. And some say: Even the sound of a woman giving birth.
What are we discussing? Based on Kli Yakkar (Bereishis 32:25) and Yismach Moshe (Bo 19:1) the Gemara can be understood as follows. We are referring to the basis for faith and understanding G-d in this world. The Romans approach this concern via their mastery of technology and science. However, you do not need to be a great philosopher or scientist, simply looking at the Sun and seeing the wonders of creation is enough. This is what the Gemara means when it compares and equates the bustle of the crowds of Rome to the Noise of the Sun travelling via its sphere. (The ancients believed that the various heavenly bodies were attached to transparent spheres. This allowed them to understand and calculate the various Orbits. In addition, since these sphere were obviously large, like any giant machinery, they believed they made noise as they rumbled about. They assumed this to be the background noise you hear ringing in your ear.)
However, that is not the end of the discussion. The imperatives of life and death itself engender a need to believe, otherwise how does all of life’s travails and successes make any sense. There are no atheists in foxholes. This is what is meant by “the sound of the soul leaving the body”. The recognition that life ends, often in a rather undignified manner. There must be more beyond all the petty strivings. More optimistically, seeing the miracle of life and birth, the sound of a woman giving birth, leads us to intuitively know that there has to be more to this life, because it is so painfully beautiful and tragic at the same time. In the end, we know the truth.
Translations Courtesy of Sefaria