When it was first formed, the air was similar to the gases of a volcano. Volcanic gases are mostly water vapor, with a little carbon dioxide, nitrogen, and chlorine. The gases were transformed into the air we breathe today when a lot of water vapor fell on the earth in the form of rain.
This rain was so intense that it created an ocean on the earth.
Life was born in the oceans, which gradually migrated from the oceans to the land, and finally became green plants.
Plants work by grinding carbon dioxide and converting it into oxygen. Thus, the carbon dioxide in the volcanic gases was gradually converted into oxygen.
This is how air as we know it today was created.

This atmosphere stays around the earth because it is attracted by the earth's gravitational pull. When the moon was formed, a large amount of gas must have been released from the rocks, but because the moon's gravitational force was too weak, the gas diffused into space and could not create an atmosphere.