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.