.Part I: Background
. Part II: Beginning; Egypt
. Part III: Syria and Jordan
. Part IV: Aftermath and Consequences
The War essentially started May 22, 1967 when Egypt announced the blockade of the straits of Tiran. That night, General Gamel Abdel Nasser is quoted as saying; "The Strait of Tiran is part of our territorial waters. No Israeli ship will ever navigate it again." The United States saw this only as clear intent to provoke a war, whereas the state of Israel, following hundreds of years of military history, catalogues this move as an actual act of war. Lyndon Johnson was later quoted as saying "If a single act of folly was more responsible for this explosion than any other it was the arbitrary and dangerous announced decision that the Straits of Tiran would be closed."
(Some would argue that when Egypt kicked out the United Nations Emergency Force from their position between the two countries a week earlier was the beginning of the war. However, it seems obvious that the war's inevitability and its start are two different questions)
Once Israel's sovereign right to use international waters was stopped by this blockade by Egypt, and their gasoline supplies were getting dangerously low, due to the aforementioned blockade, Israel decided to use military force. They argue that the beginning of the war could realistically date from this time, since it was the first warlike act perpetrated by either side. During the hiatus between the blockade and the outbreak of hostilities 2 weeks later, however, many things were occurring diplomatically. (In addition to the spate of terrorist attacks by Jordanian citizens and Syrians bombing agricultural settlements preceding the war.)
In the UN, messages were passing back and forth furiously between the United States and the USSR, discussing how to prevent violence. Unknown at the time was that the Soviets had played a large part in provoking Nasser into the war originally, telling him that Israel was poised to attack Syria, (which was false) that they would support him militarily if the US got involved, (which remains unclear but seems unlikely) and that they was no way he should be able to lose if he acted intelligently.
Israel, of course, was talking with its ally, the United States. The US was trying desperately to convince Israel that a preemptive strike was a bad idea, and that it would organize a multinational convoy to break to blockade. The convoy idea died due to lack of speed and, eventually, unwillingness of any countries to get involved. During this time, however, while its oil supplies were being depleted, Israel's cabinet voted twice to postpone any military action.
The rift between France and Israel dates from the preparation for the war, in which Israel asked for more arms than they were slated for currently, and with more speed, and france denied the request, and tried to stop arms shipments immediately, which would have been debilitating if they weren't so incompetent about doing so. (Arms continued to arrive until the middle of the war from france.)
Egypt was being slowly convinced by Russia that it needed to weather the first attack by Israel before counterattacking, as Russia did not want to be held accountable for starting the war, which in the long run would determine the layout of the Middle East in the future. Egypt probably would not have listened for much longer, but the beginning of the war of June 5, 1967 made the point moot.
The United States had the same concerns about Israel and the beginning of the war, but Israel was unable to stomach the idea that it would be forced to take a huge loss at he beginning of the war in order to not be labeled the aggressor. They were also concerned that when the heretofore neutral Jordan attacked, it would be able to easily split the country, 9 miles wide in the middle, effectively destroying Israel before the war even truly started.
On June 4, Jordan joined Nasser's coalition to destroy Israel, placing all of Jordan's troops under Nasser, and showing the Israeli's that war was not only inevitable, but also likely to come soon.
Israel attacked the morning of June 5, at 7:14 planes were launched at targets in Egypt, marking the official start of the six day war
This Writeup was based in large part on Michael Oren's Six Days of War: June 1967 and the Making of the Modern Middle East and the dates and quotes come from http://www.palestinefacts.org/pf_1948to1967_sixday_backgd.php and the links for additional information at the bottom of the page.