Here's how the layover program works
I've recently done a 10-hour layover in Cairo and took advantage of EgyptAir's service. I'll summarize how it works.
Once you arrive in the Cairo airport, if you have at least a 6-hour layover, EgyptAir is willing to give you a hotel room at the nearby Iberotel that likely includes a meal at the hotel depending on the time of the day. You will need to find the EgyptAir desk near the "Domestic Departures" sign. In my case, I spoke with them at about 5.45 AM. I had to leave a copy of my ticket and my passport with them for about an hour and 15 minutes before they prepared the hotel voucher that would entitle me to free use. This arrangement involves leaving your passport at the airport with these EgyptAir employees so that you will not need a transit visa or a regular travelers' visa to enter the country (all of this is not explained beforehand by the employees there, they assume you know everything).
Once everyone's vouchers are ready, an employee escorts you out of the airport (a police officer checks to see that your passport photo matches your appearance, and then the passport stays with the EgyptAir employees). Outside the airport, a van takes you to Iberotel along with other passengers.
Once at the Iberotel (around 7.20 AM in this case), you wait at least half an hour for them to give you your room. You get a breakfast and lunch voucher if you have time to take both of these meals at the hotel before your next flight. After your breakfast, you check into your room. AFter that, you are free to enjoy the hotel's facilities, or arrange a tour with Karnak Travel who has an office inside the hotel. An interesting note is that you can also directly arrange your tour with Karnak Travel before exiting the Cairo Airport, but that will probably mean foregoing the free hotel room at the Iberotel (there is not enough time for both EgyptAir to prepare your hotel voucher while Karnak Travel separately prepares your paperwork to enter the country without a visa, you'd need to choose one of thse options at the airport. If you have only a 5-hour layover, you may opt to deal with Karnak Travel directly at the airport to arrange a Pyramid tour). The time needed for a Pyramid-area tour is about 4 hours from starting point to coming back to the airport. There are longer tours on offer (combining Pyramids with the National Museum would be at least 6 hours) so you can choose these based on how much spare time you have.
In my case, Karnak Travel's office at the Iberotel arranged a guide and a car which I shared with a family who did the same tour. I have to say that as this was the month of Ramadan (fasting period) working hours were more limited, so I waited until their office opened at 9.30 AM to ask about their options. By 10.10 AM, we were off to the Pyramids and the nearby Sphinx (cost: 35 USD per person), but there was not too much time to see the details or talk about the region's history with the guide. It's more of a photo-taking tour than an in-depth tour of the area. In any case, because Karnak Travel is a subsidiary of EgyptAir, you can expect to be treated very well by the guide and your driver.
The tour guide is instructed to bring you back to the airport 2.5 hours before your flight, so make sure you have enough time to do the 4 hour tour polus the 2.5 hours safety margin before your next flight before you join this tour. If your time is short, you may ask the taxi desk at the Iberotel who can take you to the Pyramids and bring you back more rapidly. This way, you do not have to spend an hour at the end of your tour at the Cleopatra perfumery (selling natural perfumes) listening to their sales pitch and being expected to buy one of their 25-Euros per 2.5 ounce bottle of fragrances (still, this was fun).
At the end of the tour, you can asked to be dropped off directly at the airport without returning to the hotel (in case you checked out before leaving for the tour). You will be asked to pay a 2 USD supplement for this service. Once you are back at the airport, ask about the nearby EgyptAir service desk, where your passport and your new boarding pass for the next flight will be waiting for you. Your baggage will have been placed on your next flight in the meantime. There will be a small Arabic-language stamp in your passport at the end of the tour.
Overall, the whole thing was a bit stressful because no one from EgyptAir explained anything throughout the process, but Karnak Travel was helpful enough to sort out the Pyramids visit.