When the Bais HaMikdash stood, the Kohen Gadol was the messenger of the people who pleaded forgiveness for his nation. Today we are missing those grand ceremonies, and prayer takes its place. Therefore, as part of the Yom Kippur service, there is a lengthy and detailed description of the Avodah as conducted in the times of the Bais Hamikdosh. In fact, we act out some parts of the service, such as bowing.
Once during the recital of Aleinu and three times during the description of the Temple service we bow down to the floor. When we reach the point in the narrative where the Kohanim and the people in the courtyard of the Temple prostrate themselves on the floor upon hearing the glorious Name of G-d emanating from the mouth of the Kohen Gadol, we too bow down on the floor to Hashem.
Note that the Torah forbids prostrating oneself completely on a floor of stone. The Sages forbade complete prostration even on non-stone floors and even kneeling (which is partial prostration) on stone floors. Therefore, if the floor of the synagogue is made of stone one must put down something before bowing. While this doesn’t apply to non-stone floors, there are customs to put something down regardless.