Israel won the Six-Day War due to many reasons, the first was that the Israelis took the Arab countries by surprise, they launched preemptive attacks which took out the Arab air forces while the planes were still grounded.
Another reason is that the IDF are highly motivated as they know failure is not an option because then Israel will cease to exist. The Arab countries at the time were struggling to stop their soldiers from defecting.
The IDF are some of the world's top combat specialist and are highly trained and skilled in the art of warfare. This did not help the Arab nations as their soldiers did not know how to us the high tech weaponry the soviets supplied them with.
Lastly, the Israelis were organised and were able to coordinate well planned and organised attacks, while the Arab nations could not organise their own troops or attacks.
