How did Israel win the Six-Day War?

One point which has not been mentioned yet... it has been alleged that Soviet industry provided export versions of hardware (tanks, etc) referred to as 'monkey models' which were inferior to the variants issued to Soviet troops proper.

The first use of the term is credited to Vladimir Bogdanovich Rezun (pen name 'Viktor Suvorov'), a Soviet intelligence officer who defected to the west. He mentions this in his 1982 book 'Inside the Soviet Army'.

Soviet export models were simpler for many reasons, principally: to reduce cost, to increase mass production, to provide western intelligence with disinformation on Soviet capability, and to enable Soviet troops to have a qualitative edge against their allies if for whatever reason they were suddenly no longer allies.

So the Soviet military hardware the Arabs had wasn't as good as it could have been. Furthermore, Suvorov took a dim view of the quality of his own junior officers. And he pointed out that the military advisors sent to help the Arabs as they prepared for the war were, like the tanks they'd exported, not the highest possible quality available. Consequently he recalled many colleagues being convinced that the Arabs should win, because of a purely numerical assessment, instead of considering the balance of relative quality.

EDIT: Years later, after first writing this answer, it appears that Suvorov was more accurate than many assumed. At first, Western analysts assumed Suvorov was making it all up, because his conclusion was that the entire Soviet military was hopelessly corrupt, and that most junior and senior officers suffered delusional optimism. Suvorov told one peer in the 1960s that he doubted the Red Army could defeat West Germany, never mind combined NATO forces.

His first book, published in 1981, followed his defection to the UK in 1978. But this conflicted with a western narrative, pushed especially by the Reagan administration, that the Soviets were a serious threat. The Soviet government and CIA both knew by the early 1980s that Soviet society was in seemingly terminal decline. But this professional assessment was ignored by Reagan's administration for political reasons.

Russia's invasion of Ukraine in early 2022 suffered from the same problems Suvorov claimed existed in 1981. Principally corruption, which created equipment shortages and low morale. In one interview with Der Spiegel, Ukraine's head of military intelligence, Kyrylo Budanov, explained his thoughts on Russia's military failures:

"I am amazed at how incompetently and carelessly Russian commanders have approached such a major operation. If they really believed that they could do it in three days - and, as far as we know, they absolutely believed that they could - then the Russian leadership is probably wondering whether their generals are competent at all. They took their wishful thinking for reality [...] We asked ourselves this question. And we found only one answer: the extremely low professional level of their generals. One of the reasons for this is nepotism in the Russian army, sham generals, relatives of some officials who don’t meet the performance criteria, unready to solve tasks."

The American government failed to acknowledge the decline and demise of the USSR, despite their best experts telling them the truth. Melvin Goodman was the CIA's Head of the Office of Soviet Affairs 1976-87, and he concluded:

“I think probably one of the greatest myths in America, in the political discourse now, is that actions of the American government were responsible for the collapse of the Soviet Union. 

“The Soviet Union collapsed like a house of cards because it was a house of cards. It rotted away from within. The economy was rotten, the political process was rotten, they had developed a central government that was no longer believed by people outside of Moscow, there was total cynicism throughout the Soviet system of governance, there was no real civil society. 

“But the Reagan administration and their minions will tell you that Afghanistan led to the collapse of the Soviet Union itself, the collapse of the Berlin Wall in 1989, the collapse of the East European empire. We were saying that this was entirely fanciful. And the United States missed all of this, because they believed their own myths and their own fanciful notions. They had become their own victims of their own lies.”

The American government also got the Falklands War wrong, believing the British could not retake the islands. More recently, the American government failed to anticipate how the war in Ukraine would unfold, delusional in their belief that the Russians would take Ukraine in a few days. Seems like Suvorov's writing is worth taking into account, because it would have accurately predicted the evolution of the Russian invasion of Ukraine.

The United States did not assist Israel in the six days war. Israel was not a client state of the United States before the six days war. The IDF did not fly American airplanes for example. The IDF flew French Mirages in 67. The special relationship between Israel and the US with regard to financial aid and military aid started after the six day war. In 1967 total US aid to Israel was $23.7 million dollars. In 1968 it would quadruple $106.5, in 1971 that number would increase again by 5 times or $634 million dollars. By 1974 total US aid was up to 2.6 billion. This is the period when the special relationship between the US and Israel really began.

As for how the Israeli's won the six day war.

• They won because they attacked first and caught the larger arguable better equipped Egyptian air force on the ground and destroyed it.(*)
  Operation Focus. On June 5th 1967, (Egyptian losses: 338 aircraft and 100 pilots; Israeli losses: 19 aircraft). That set the stage for everything which came after. It ensured that Israel would have air superiority though out the conflict

• Poor Leadership on the part of the Arabs. Egypt had a lot of forces in the Sinai. When Israel plastered their airforce, airfields and air defense Egyptian forces panicked and tried to retreat back into Egypt. It was disorganized and Israel made them pay for it dearly with their newly won air superiority. Syria and Jordan
  listening to Egyptian radio reports believed Egypt was winning, and thought they were attacking Israel when she was vulnerable. They were mistaken and both of their air forces were either destroyed(Jordan) or decimated(Syria) just as quickly as Egypt's had been dispatched.

• You'd also have to add professionalism of the IDF to the mix.

Israel's land forces grabbed Sinai, Golan Heights, Jerusalem and the West Bank from Egypt, Jordan and Syria; increasing the size of Israel by a factor of 3.

Why did Israel attack Egypt first. Egypt and Israel had been fighting boarder skirmishes for years, so tensions were already high. Also for years the Egyptians had been denying use of the Suez Canal to Israeli shipping. But the cases belli of the attack was Egypt blockading the Strait of Tiran, Israel's only port on the Red Sea; The of Eilat. This blockade meant all of Israel's oil ( only the Shah's Iran would sell oil to Israel at this time ) would have to travel around the horn of Africa which was deemed an existential threat. Israel's leadership felt they had to respond.

(*) 1967 was before the F-15. The United States was just starting to fly the McDonnell Douglas F-4 Phantom which wasn't a superior aircraft to the Soviet Mig 21. Certainly it didn't outclass the Mig as the now retired F-15 outclassed it's contemporaries.

605 F-4 Phantom aircraft were lost in combat in Vietnam, compared to 100 MiG-17's and 86 Mig-21's. (includes air to air, and ground to air losses) That's an astounding number when you consider no F-15 in 20 years of service was ever lost to air to air combat. Not by the United States nor any country which flew them.

Anyway Israel wasn't even flying the Phantom 1967, the IDF wouldn't receive the phantom until 1969. Israel was flying the French Mirage against Egyptian and Syrian Migs.. Arguable the Mirage's were inferior planes and both Egypt and Syria had numeric superiority. The professionalism and training of the IDF to take maximum advantage of the surprise attack was also a difference maker.

The answer to how Israel won the Six Day War has filled numerous books and cannot be done justice in a short-form resource like this. At best, one can only give a summary in this space, but even the summaries that came before mine have given short-shrift to some key factors.

Obviously, air superiority ranks first. The Israeli Air Force's success at catching the Egyptian and Syrian air forces on the ground paved the way for full IDF control of the skies thereafter. That success came partly through planning and largely through Egypt's failure to have hardened hangers for its aircraft, adequate air defense capabilities, or better communications and control systems. Moreover, their reliance on Soviet military doctrine left very little initiative with local commanders, meaning that a shutdown of communications would keep Arab forces with their heads down. It has been rumored that Israel effectively intercepted Arab communications and made its own broadcasts pretending to be Arab leaders, giving out false and misleading orders.

Critical intelligence that came from human intelligence networks, such as Eli Cohen, and more conventional electronic intelligence (i.e. monitoring various aspects of enemy communications whether or not codes had been broken) were even more valuable to Israel. Cohen, among other things, was said to have posed as a trusted advisor to a Syrian general and made the comment that cedar trees should be planted by Syrian positions on the Golan Heights so as to provide shade to the soldiers there. The trees also provided perfect positioning points for Israeli artillery. According to Isser Harel's book, Israel also used humint from Cohen to find out what Arab soldiers were fed by their own armies; Israel made sure that it would feed POWs better and more food than they received in their own camps with the idea that the Arab fighters would feel demoralized with the knowledge that Israel had more food to feed prisoners than their own armies had to feed their soldiers.

Jordan's lateness in joining the battle allowed Israel to focus on its enemies north and south first, and after taking Sinai and the Golan Heights, Israel could focus on the West Bank without worry about its flanks. Much credit for that must be given to Egypt's Nasser who incorrectly boasted to Jordan's king that Egypt had destroyed much of the IDF's air power.

Nasser's ineptitude cannot be limited to that one incident. Prior to the war, Nasser changed his mind four times on his country's invasion strategy and goals, forcing large troop movements multiple times, thereby creating a sense of chaos among the troops, and sucking on factors such as morale and readiness.

The Egyptians, along with Syria and Jordan, had worked up a plan to attack Israel a couple of weeks prior to the actual war. Apparently some recently declassified documents confirm that the Egyptians had planned to launch bombers against Israel to take out their airfields and other strategic military positions. They had also deployed a number of tank units that were prepared to slice through the country.

The Israelis were trying to convince Lyndon Johnson to provide support, but the US was already involved with VietNam and wasn't prepared to provide any type of military assistance in defending Israel. However. the Israelis did convince him that an attack was imminent. As a result, Johnson warned Egypt not to attack and informed the Soviet Union that if Egypt did attack, the US would hold the Russians accountable. The combined pressure from the US and Russia proved to be enough to convince Egypt to postpone their attack.

About a week later, Israel decided that the Arab countries were not going to delay much longer, so they chose to take the offensive. The Israeli air force wiped out the Egyptian and Syrian air forces while they were still on the ground, and therefore had no resistance at all in the air. Two days later the Israelis destroyed almost all of the Egyptian tanks, thereby eliminating the last opportunity the Arab countries had to mount any form of attack.

Basically, the Israelis took the initiative away from Egypt and then turned it around and did exactly what the Egyptians were planning to do to Israel. The only form of assistance that Israel received was the political intervention leading up to the actual event, and that proved to be enough.