3 Steps taken prior to and follwing the Amidah

Before the Amidah, the custom is take three steps back and then three steps forward. The idea is to step into a place of prayer, a different headspace that is more than a few inches distant from where you were, in which you know that you are standing before the Creator.

So too, at the end of the Amidah, we take three steps back and bow, as we leave the space of prayer. Rambam writes that you should take leave from your prayer, the way you take leave of a ruler. So too, the priests would take three steps back in withdrawing from the altar. In this spirit, some people also walk backwards off the bimah, out of synagogue, or away from the Western Wall because you would not show your back to a king.

There are different interpretations of the three steps; Orach Hashulchan says it alludes to the three-fold “hoshech anan va’arafel” darkness, cloud and obscurity that covered Mount Sinai when Moses received the Torah. The idea is to step back from a space of spiritual intensity and return to the everyday world.

At the end of the Amidah we take 3 steps backwards as mentioned in the Gemara (Yoma 53a) as the proper way to take leave of a king. 

There is another deeper reason for taking three steps backwards, but the simple explanation for the custom is in order to recall the three miles which the Jewish people retreated from Mount Sinai, (from fear, upon hearing G-d's voice), before they returned and were given the Torah.