The simple answer from the perspective of Operational Warfare*: The Israeli Defense Force was simply better than their Arab opponents in '67, though certainly not infallible.

The Israeli Army of 1967 followed a frightening operational doctrine: it took a very Germanic obsession with maneuver warfare, individual initiative, and ceaseless aggression, turned that obsession up to an unbelievable extent, and then garnished it with the very American obsession of deploying obscene amounts of firepower from the get-go.

Israeli doctrine evolved along the same lines as German doctrine for the same reasons - both nations were almost completely surrounded by forces much larger and more powerful than theirs that would win in any straight up slug-fest. As a result, both the German military and the IDF sought to win decisive engagements against forces much, much larger than them, primarily through maneuvering multiple independent forces with the objective of utterly confounding and destroying the enemy by falling upon their flanks and their rear faster than they could react.

The one primary difference between German and Israeli doctrine was influenced by Moshe Dayan's chance meeting with a US Army officer, Abraham Baum. In the later stages of World War II, Baum had led a task force of lightly armored vehicles on a daring raid to try and liberate an American POW camp near Hammelburg. The most important lesson Baum had taken away from the experience was that modern warfare was dictated in large part by the sheer shock value of fire-superiority. His advice to Dayan was:

"I don't know what you're doing in Israel, but I suggest that the moment you come to an obstacle, you open fire in all directions with whatever you have [...] You should shoot and drive, shoot and drive."

This peculiar doctrine was not without its faults: Israeli units were highly undisciplined in combat - an officer who rejected his superiors' orders and prosecuted independent action based on local information was often revered and lauded... so long as he succeeded. Israeli forces had very poor logistical support - in Soviet terms, they could not perform "consecutive operations" in the slightest (keep in mind, this wasn't seen as a problem - in the minds of IDF strategists, if they ever lost one battle, then that was it; there was no point in planning for a second battle when Israel would already be wiped from the map). Finally, Israeli forces were very, very prideful and convinced that they could prevail over any Arab army because they were better motivated, better trained, and had more will to win (this didn't play a role in the Six Days, but it would bite back with a vengeance six years later, when their pride led them into foolhardy frontal assaults against fortified positions).

On the opposite side of the border, the Soviet client-states arrayed against Israel were far more generously equipped with export T-54's and -55's and all the requisite logistical support such armored divisions required. Egypt, in particular, had an army with a TO&E of sobering proportions for the Middle East. They were not, however, properly schooled in Soviet warfare. As Dayan wrote of the forces he faced in '56:

"We should avoid analogies whereby Egyptian units would be expected to behave as European armies would in similar circumstances"

And he was right, both in '56 and '67. Which brings us to your question. Why did the Israeli's win so thoroughly in '67?

Much is made in 1967 of the preemptive airstrike against the Egyptian air-force, and well it should. There was, however, significant fighting on the ground that cannot be ignored. In the first day of combat, the IDF sent three division-strength units (called ugdah) into the fray against twice their number in soundly-deployed Egyptian units, and thoroughly won. General Israel Tal's division chewed its way through tough fighting in the north, finally battering aside the Egyptians with the help of IAF air support and an unimaginable amount of on-the-spot improvisation by Israeli commanders. The late General Ariel Sharon fought a massive set-piece battle in the south, soundly defeating the Egyptian defenders in a complicated series of artillery-supported assaults, tank maneuvers, and flanking attacks. In between, General Avraham Yoffe's unit found itself sitting astride an Egyptian advance into thoroughly horrible terrain, and the Egyptians simply dashed themselves to pieces against them. These initial actions simply broke Egyptian units everywhere, as their poorly-trained and low-morale soldiers were overwhelmed by Israeli firepower and the tendency of Israeli units to appear out of the dark from unexpected angles at the worst possible times.

Starting on the second day of the offensive, Israeli forces were streaming across the Sinai in ragged columns. In the end, the Israeli advance overran and overwhelmed a demoralized Egyptian army on the retreat through the Sinai. This retreat ended at the Mitla and Jiddi passes, where fleeing Egyptian units attempted (and failed) to run a gauntlet of Israeli blocking positions and were slaughtered as they approached.

What is often glossed over is just how close Israeli efforts were to failing many times, though never catastrophically. Unit cohesion was difficult to maintain on the drive across the Sinai, intermingled as Israeli units were with retreating Egyptians. Even worse, logistical support was never Israel's strong suit, and tank after tank was sidelined during that drive by something so pedestrian as running out of fuel. Of one entire brigade, a mere nine tanks were able to finish the drive to Mitla and Jiddi!

There also is some glaring negative space in the overview of the Six Day War: The Israeli take-away from their campaign across the Sinai was that tanks and aircraft working in concert was the way of the future. They had, in fact, already begun de-emphasizing infantry training in favor of throwing all their ground eggs into an armored basket. The high levels of individual initiative practiced by Israeli forces were beneficial in overcoming specific battlefield problems, but highly detrimental to maintaining unit cohesion and logistics. Finally, Israeli armed forces came out of the Six Day War convinced in no small part that their Arab enemies were inferior soldiers (mostly accurate) inferior people (debatable and obviously culturally biased) and would never amount to a serious threat (woefully wrong).

All this mixed together into a perfect storm of pride and disaster in 1973, but that's another story for another time.

*Much of this argument is laid out in Blitzkrieg to Desert Storm: The Evolution of Operational Warfare

By making a surprise attack with amazingly good timing and good tactics. The attack marked the combat debut of the anti-runway Durandel bomb, which helped them trap MiGs on the ground to be strafed and rocketed, and every single Egyptian anti-aircraft battery and SAM site was under "no fire" orders because of a scheduled aerial visit by the Egyptian minister of defense; only a single battery violated this order and fired upon the Israeli attackers. The initial ground assault upon Egypt received similar such luck: Egyptians misidentified Israeli troops as friendlies and also identified the main attack as a diversion (similar to Germany misidentifying Overlord as a diversion from a landing in Calais).

Source is the book "Six Days of War: June 1967 and the Making of the Modern Middle East" by Michael B. Oren