As a North African Arab (not Egyptian), I interact with other arabs each using their own dialect, because they wouldn't be able to understand my dialect easily. However, the claim being made about Modern Standard (not classical) Arabic or MSA for short, is that you can use it to communicate with any Arab around the world, anyone with highschool or even middle school level of education can communicate with you.
You may hear the claim that Egyptian Arabic is understood by everyone in the Arab world and that's true. But reminder that Egyptians understand most other dialects. You can imagine a person speaking Qatari with an Egyptian, it's possible that each uses their dialect and understand each other. But that wouldn't be the case for someone that only knows Egyptian.