It means that you have embarrassed yourself at the highest elevated levels due to you being absolutely incorrect about something in particular, yet you continuously argued your point adamantly with pride attached to your ego because of only you being so sure and full of yourself that you are definitely right, until you’re proven totally wrong!
You and your debating usually occurs in the midst of a group or crowd of others in tuned, with all others knowing or agreeing that you are wrong as well. They will either remain silent until the revealing outcome takes place, or all others will speak up on the matter in efforts to keep you from embarrassing yourself.
Yet, it’s too late to save you being shamed because you’ve allowed your pride and ego to take over you, misinforming you that everyone else will suffer the embarrassment instead.
Following the fact that you were wrong, and due to you so aggressively, so self-assuredly, marked by a pompously egotistical attitude of you arguing your point that you were right, is when someone will say to you,
“Now wipe the egg off of your face,” or,
“How does it feel to have egg all over your face?” or
say to someone else, “Look at them. Left with an egg on their face!” or
some other clever phrase coined with having a raw egg cracked on the forehead, releasing its slimy texture to drip and run down the face.
This can also occur in a one-on-one setting as well, without others around, only between two individuals.