After the Lord and the two men who came with Him finished eating, they arose and looked toward Sodom. The Lord said He had come down to examine the deeds of the inhabitants of Sodom and Gomorrah (vs16-21). The men went to Sodom, while Abraham still stood before the Lord. (A similar story is documented in Genesis 11:5.)
After the two men departed, the famous negotiations began between the Lord and Abraham over saving the righteous of Sodom. These end with the verse, "So the LORD went His way as soon as He had finished speaking with Abraham; and Abraham returned to his place" (vs33).
The two men who left Abraham and went to Sodom were angels sent to save Lot and his family from the destruction the Lord was about to rain on the sinful cities of Sodom and Gomorrah. The word angel does not always describe a winged image; it usually refers to a messenger.
Unfortunately, rabbinical Judaism does not take a Scriptural stance regarding the identity of Abraham's important visitor. It maintains the position that all three men who visited Abraham were angels.
This explanation does not follow God's Word. The Word teaches clearly that El Shaddai, the godly entity whom people are allowed to see, appeared to Abraham in the form of a man. Readers must decide to whom they should listen: the wisdom of men or the explicit Word of God.