It’s super easy.
Go to Traffic Administration in Nasr City (Arabic: Edaret Elmorour El3ama or ادارة المرور العامة) that is right next to Mosque of Rabea elAdaweya (Arabic: جامع رابعة العدوية) and issue a slip of paper from them that you take to Nadi Assayarat in downtown Cairo (ATCE. see below). When you go to Edaret Elmorour El3ama just tell them you want the paper for the international driving license. All you need with you is your normal license and no fees paid. It all takes few minutes on one of their windows at the ground floor. Parking there is not bad. They close on 2pm in normal days and 1.30pm (usually means they close 1pm) on Ramadan.
Please note that you will not be granted the slip of paper unless your current driving license is valid for at least one year. If that is not the case you will be required to renew your driving license first.
Then go to Automobile and Touring Club of Egypt (Arabic: Nadi Assayarat نادى السيارات) www.atce-egypt.org (Phone 25743355) in downtown. Parking there is bad during the day so you may need to leave the car somewhere else and use Taxi. They require one picture, the slip of paper from Edaret Elmorour El3ama in Madainat Nasr, your Egyptian driver’s license, your ID and a copy of your ID, and 40 LE. Your international license will be ready within 2-3 hours. The international license is valid for one year.
When abroad you must keep your original Driver's License. The International Driver's License is supposed to be a translation so you show both when necessary.