You're walking into a building, and there's someone walking behind you. You open the door, and you hold it so that the person can go through first. You want to tell that person to go through before you, so you say this.
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This is a polite phrase that you say to tell someone that they can go before you. Use it when you're holding a door open for someone, or when you're in a line for something and want to let another person go before you.
Sometimes, two people will compete to allow the other person to go first:
A: After you.
B: No, after you.