Part I: Background
. Part II: The Beginning; Egypt
. Part III: Syria and Jordan
. Part IV: Aftermath and Consequences
Fighting began on the morning of June 5, 1967, at 7:14 Israel launched its entire Air Force (excepting 14 Mirages, which were defending Israeli Airspace) at Egypt, 183 planes, in the most striking example of a pre-emptive attack in modern military history. The timing of the strike, according to Israeli sources, was chosen to attack the Egyptian airfields while the pilots were at breakfast, either for humanitarian purposes, or simply to make sure they were not at the planes and able to take off. The concentration of the initial attack was on Egyptian airfields, and flying into Egyptian airspace below the radar, Israel managed to enter almost completely undetected, destroying the runways (and trapping the planes) of all of Egypt's vaunted air force in one blow.
A second wave was launched at Egypt, after refeuling, at 9:34, and hit more Egyptian airports, and several radar installations. The third wave, at 12:45, attacked Jordan's air force, which was completely destroyed, and Syria's air force, which was very severely damaged. One airport in Iraq was also attacked, and the more were destroyed the next day. Losses for the first day of air combat were approximately 70% of the Arabs combined air power, and about 10% of Israel's.
During the first half hour of the air strike, Israeli forces in the south got ready, and invaded the Sinai peninsula. This was a political decision made by the cabinet, and was intended for use a material to trade with Egypt after the war. Israel skirted the gaza strip at first, capturing it several days later.
Egypt's deployment in Sinai Peninsula at the time was a complete mess. General Gammel Abdel Nasser, the country's ruler, was having an internecine fight with his main generals. The three tiered defence that was supposed to be in place for the defence of the Sinai was being moved in multiple direction simultaneously just as Israel invaded.
As Israeli troops moved forward, they met a split between almost no resistance, and very fierce resistance. In many key strategic locations, Egyptian troops abandoned bases, material, tanks, and artillery pieces to the Israelis. Official Israeli counts were 625 tanks, 200 usable, 750 Artillery pieces, and 2,500 trucks and other personnel carriers. In other locations, including two key bases along the invasion route, there was fierce fighting that left a total of 338 Israeli troops dead, 69 tanks damaged and 63 tanks destroyed. This compared to total Egyptian losses of 19,500, which includes 4,500 captured.
Adding to the confusion, because of the previously mentioned split between Nasser and his military, Nasser was getting his news from Egyptian propaganda sources, which were not in contact with the actual troops. He believed that Egyptian forces were almost in Tel Aviv, and that the Air force was untouched, and destroying Israel rapidly. This led to his rejecting any notion in the United Nations for a cease fire, and destroyed any chance for Egypt to cut it's losses.
During this time, Israeli politicians were attempting to keep up with troop movements, and at several times the cabinet asked for key positions to either be taken or bypassed, and they were already well behind the line of advancing troops. The overall goal was to cature enough land to be able to negotiate a favorable deal with Egypt, but they eventually reached the Suez Canal, and Israeli politicians absolutely forbid troops from crossing, as they did not want to be invading Egypt's heartland.
Some troops waited at the Suez Canal for several days (others were moved to other locales) while the war with Jordan and Syria was concluded.
I realize that this was written almost entirely from a Israeli perspective. This is clearly partially a bias, but writing from any other perspective is almost impossible, given the fact that their countries allow no access to their archives, and almost nothing is known about how the leadership reacter or how the decisions they made were reached.