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][1] million dollars.  In 1968 it would quadruple [$106.5][1],  in 1971 that number would increase again by 5 times or [$634 million dollars][1].  By 1974 total US aid was up to [2.6 billion][1].  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][2]. 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][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][4] of the attack was Egypt blockading the  Strait of Tiran, blockading Israel's port of Eilat.  This was Israel's only Red Sea port and it's blockade meant all her oil ( from the shah's Iran )  would have to travel around the horn of Africa which was an existential threat Israel had to respond too.  


(*) 1967 was before the F-15.  The United States was just starting to fly flying 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 action was ever lost to air to air combat.

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.  The professionalism and training of the IDF was the difference maker. 


[1]:http://www.jewishvirtuallibrary.org/total-u-s-foreign-aid-to-israel-1949-present
[2]:https://en.wikipedia.org/wiki/Operation_Focus
[3]:http://www.newworldencyclopedia.org/entry/Six-Day_War
[4]:http://www.sixdaywar.org/content/israel.asp