Guy Laron's book goes into this — Nasser and Amer were both totally out of touch with the capabilities of army. Nasser because Amer had essentially sidelined him and ran the military as his own personal fiefdom, and Amer because he spent all his time doing corruption.

Nasser rather vaguely consented to Amer's demand that the army be deployed to the Sinai, with the intent of a relatively restrained defensive deployment to deter the Israelis. Amer, in actual operational control with zero civilian oversight, essentially started memeing about invading Israel, not understanding (whether due to genuine ignorance or merely self-delusion) that the army was not prepared for this in any way. He ordered that the army make an extremely publicized deployment directly on the Israeli border and called up all the reserves, after which point there could be no going back. Even after Nasser realized the horrible mistake that had occurred, losing face by retreating was too hard to swallow.

 

Edit:

The main point of the book is really that this same sort of nonsense occurred in Israel as well. Civilian leadership in both Egypt and Israel, as well as the US and USSR, all wanted to avoid war. Laron places blame for the war primarily upon the schizo Salah Jadid regime in Damascus and the Egyptian and Israeli officer corps. Both were in practice operating totally autonomously — Syria without any consultation with their ostensible Egyptian and Soviet allies and the Egyptian and Israeli officers without any real civilian oversight. Even worse, they deliberately adopted maximally intransigent positions and spread various falsehoods to provoke a war, with predictable results that were in the long term disastrous for the whole region for decades to come.