Israel didn't win because it overpowered its enemies and muscled them to the ground. If Israel had had that kind of power, the Arabs wouldn't have made their plans to attack. The only reason Nasser was able to convince the Syrians and Jordanians to participate is that the Arabs all thought they had military superiority and would win easily.

Faced with Arab mobilization and clear intelligence that they planned to attack, Israel won because it made a sneak attack and caught the Arabs by surprise. Specifically, the Israelis ordered an air strike against Egypt's air assets on the ground, which was spectacularly successful.

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There was one way in which Israel was indeed vastly superior: it had an incalculably better officer and noncomm corps. For instance, analysis of the battle in the Golan Heights shows that Israel won in early fighting with an inferior armored force because the Syrians used awesomely bad tactics which permitted the Israelis to pick their tanks off easily while losing few of their own. The Israeli tankers used cover well, and coordinated movement and fire, because they had good sergeants. The Syrians were bumblers and basically threw their tanks into combat with little plan, and saw them get destroyed.

However, even to get the kind of military force needed to be competitive, Israel had to shut down its civilian economy. Its standing army wasn't even remotely large enough to fight against a threat that large. And the reserves couldn't be kept mobilized for more than a month without causing Israel to self-destruct economically. Part of why Israel jumped the gun was to gain the advantage of surprise, but part of why was that Israel had to get the war over with so that it could restart its economy.

Looking back on it, it can seem as if 1967 was an easy victory for Israel. But that belies the reality: it was a very close thing. If the Arab attack had happened according to plan, it is by no means certain that Israel would have survived.

The idea that Israel's army outnumbered the Arabs came up decades later and is simply not true.

Israel could only field a large army because she shut down her economy. And even then the army was not the same size as the standing armies of most of the Arab countries.

The only reason there was some parity on the battle field was because the Arabs were simply worse at mobilising their army and didn't risk shutting down their economies. But to claim that Israel outnumbered them is like saying that someone won in a game of Chess because he "outnumbered" his opponent after he managed to develop his pieces better. (And it get's more ridiculous if the winning player actually had a handicap in the beginning, like no rooks.) That player won because he was ultimately better, not because it was clear that he would win.