So like all history, its a bit more complicated than that. First off, Syria had created a new organization called El-Fatah (not the Fatah of the PLO today), which had taken up the mantle of earlier Fedayeen raids into Northern Israel. Also, the Arab world was split in the ways it wanted to achieve success against Israel. Syria's president Al Atassi demanded a war for the liberation of Palestine be fought. King Faisal of KSA, who's main resource was oil and therefore western money, considered Syria's bloodlust to be untrustworthy. Nasser was caught in the middle, but as Arab nationalism was his guiding force, the radical feelings of Syria found his ear more often than the conservative traditions of KSA and Jordan.

For months leading up to the 6 Day War, Israel and Syria engaged in border skirmishes. El-Fatah attacks mounted in ferocity, and Israel was preparing for the impending war with Syria, which PM Levi Eshkol warned was imminent if El-Fatah attacks continued. The pressure on Nasser to help Syria grew tremendously, and on May 18 he disinvited the UNEF patrol force on the Egypt-Israel border, effectively terminating any chance of UN intervention in the upcoming war. This decision was lambasted by the major powers at the time, and the Israeli foreign minister Abba Eban stated "the umbrella was removed at the precise moment it began to rain." Israeli and Egyptian forces began to amass on their sides of the border. Finally, on May 22, Nasser closed the straits of Tiran, cutting off Israeli shipping, and blockading the port of Eilat. This was to lead directly to the war.

On June 5, Israel mounted a series of lightning attacks on the Egyptian air force, pummeling it in a move that would be regarded by military historians as one of the smartest attacks in modern history. What must be understood is that this attack gave Israel control of the air from the Sinai to the Golan Heights. This took place at 7:45 am, and the rest of the morning was spent trying to reason with Jordan not to attack, but to no avail.

At 11:50 am, Syria, Iraqi, and Jordanian air planes attacked Israel, but were driven back by Israeli forces, and the main air bases in the three countries were destroyed. This was a cause for low morale amongst the Arab coalition. All of this occurred in the first morning of the first day.

On June 7, the third day, Syria had still not crossed the border. Israeli army leaders discussed attacking the Golan Heights, but were fearful of a Soviet reprisal, and so they held back. That is, until news of the crumbling Egyptian military reached the Syrians in the Golan, who according to reports to Israeli central command, were fleeing their positions as well. On the morning of June 9, Moshe Dayan gave the order to attack the Golan, and the most severe fighting of the war broke out. Dayan and Rabin had differing views of how the Golan should be taken; Dayan wanted to cease attacking by the morning of June 10, and Rabin wanted to take the Syrian town of Quneitra, which was less than 40 miles from Damascus. The Syrians, fearing an occupation of their capital, ordered their forces to move east of Quneitra. At 8:30 am on June 10, Syrian radio announced that Quneitra had fallen, and its defenders fled to Damascus. This gave Israel complete control over the Golan excepting Mt. Hermon, which was taken two days later after the ceasefire, although the stronghold there was already deserted so no shots were fired.

What it comes down to is that the Arab coalition failed to communicate their plans with each other. After the first day Israel controlled the air, and Arab morale was low. The Arab forces put their faith in their greater numbers, not their military proficiency. Syria refused to cross the border into Israel, and paid the price by their inaction. And Syria did attack while Israel was sweeping into Sinai, it just so happened that they were beat by Israeli forces.

Sources: "Israel" by Martin Gilbert, and "Why Nations Go To War" by John Stoessinger

I'll go a bit more into the Syrian actions, but the already-up answer by /u/rambaz710 does a good job of showing some of the lead-up to the war itself, and a bit of the war. If you want a very long read on the lead-up to the war, I wrote one here.

Let's start at the beginning. The Israeli pre-emptive strike on Egypt commenced on June 5, at 7:10AM Israel-time. By 7:30AM, 200 Israeli planes were in the air. They flew low, by one account as low as 15 meters off the ground (50 feet for us Americans), to avoid radar. A large chunk of the planes went out over the Mediterranean to bank in on Egypt without being detected. There was complete radio silence, and they used hand signals to communicate. Pilots prepared to crash into the sea in the event of technical trouble, because they couldn't radio for help.

Egypt's planes, by comparison, were on the ground. They anticipated attacks at dawn, flew their sunrise patrols, and were back at base by 7:15AM Israeli time. There were four training flights in the air without weapons, and two transport planes, one of them carrying Nasser's second in command and the other carrying the Iraqi Prime Minister and a Soviet adviser. Israel was afraid the planes would detect them, but it didn't happen. In reality, the radar that detected them was Jordanian, one of the best radar facilities in the Middle East at the time. The Jordanians then radioed "war" up their chain of command, and sent it to Cairo. Cairo had changed their codes the day before without telling Jordan, and they were confused by the message. Oh, and the worst part for Egypt is, their chief decoder, their air chief, and the defense minister were all asleep. Reports did make it to air force intelligence, who sent it to the Supreme Headquarters, but those officers did not respond.

Planes on the ground were destroyed. Israeli planes dropped bombs that dug into runways, rendering them totally unusable, and leaving holes 5 meters wide and 1.6 meters deep. Anti-air forces had already been told not to fire, because Nasser's second in command was in the air and they were afraid he'd be hit. Israel lost only 8 planes, and 5 pilots. Egypt lost 286. 13 bases were inoperable, as were 23 radar and anti-air sites. The Egyptian air force suffered much larger damage than Israel could've imagined.

I say this because the air war was pivotal. It provided Israel with the ability to maintain air superiority throughout the war, and dominate Egyptian ground forces from the air. All of this happened within 3-4 hours. The ground attack was already given the go-ahead during this time, and by 8:15AM tanks started crossing the border. They were momentarily stopped at Umm Qatef by enemy fire, but by dusk the Israelis were firing to soften up the target, and ready to attack.

So, where was Syria? They activated almost immediately, though it took them a few hours due to a training exercise and needing to then get back to it. By noon on June 5, they were attacking Israeli communities in the Galilee, in Israel's north. They sent 12 MiGs. Three were shot down, and the rest were driven off by Israeli fighters that intercepted. Thing is, most of Syria's air force was eliminated early in the day. The MiGs attacking at noon were driven off, but followed up by Israel's air force (having refueled and rearmed after hitting Egypt) attacking Syria at 12:45PM. This assault, the third attack (following two waves hitting Egypt) destroyed every Jordanian aircraft (28 planes), and half of Syria's aircraft (53 planes). The Israelis also destroyed 10 Iraqi aircraft who were attacking Israel.

Syria and Jordan were Israel's second target, and they did indeed mobilize air forces to attempt to attack Israeli positions. They were simply grossly outmatched, poorly trained, and poorly coordinated.

Israel swept into the Sinai by June 6, with the general retreat order that was given to Egyptian forces helping that along. The Jordanians had already begun shelling Jerusalem and the war was on, and the Syrians had intermittently shelled Israeli communities in the north. Yet they ultimately never attacked. Why? Well, they were told to wait until 5:45AM on June 6. They began the shelling with the intent of softening up the targets at 2:30PM on June 5. It looked like Syria's forces were afraid to even advance. Syria's 17th Mechanized Brigade made it to the border, and stopped. They claimed they had to conduct recon first, and then said they didn't have any commands to move up. What happened to the Iraqi 8th Brigade was perhaps instructive: they attempted to cross the Damiya Bridge (over the Jordan river) and were bombarded by Israeli planes and decimated.

Israeli forces were unsure if Syria was brave enough to start a land assault, particularly after the punishing beating their air force took. So long as Syria stayed out of the land war and kept it to shelling, Israel was content to shell back and leave it at that. But at 2:00AM on June 6, the second day of the war, Syria began bombarding Israeli positions more heavily than before. Syria, which had been demoralized by the air force losses, suddenly regained confidence when Israel did not respond to the shelling. Syria launched its land war at 7:00AM. They feinted and went towards Kibbutz Dan in the north, seeing that it was completely deserted (or so they thought; the inhabitants were simply hiding in bomb shelters). Then the alarm sounded, and the inhabitants ran to the perimeter, firing at Syrian forces and catching them by surprise. The main Syrian thrust, near the Sea of Galilee, never ended up coming into reality. The three brigades sent were unfamiliar with the terrain and failed to arrive at the launching site. The tanks couldn't cross the narrow bridges over the Jordan river, and lacked radio contact with infantry. Some troops simply refused to move. Then the Israeli air and artillery assaults began, leading anyone who had started to attack to head right back to base. By the evening, most of the unprepared and untested Syrian forces had wandered back to base, or been killed. Syria stuck to a defensive plan instead, leading Israel's Yitzhak Rabin (later Israel's Prime Minister, and at the time serving as the Israeli Defense Force's Chief of Staff) to comment, "Syria is willing to fight to the last Egyptian".

So the Syrians did attack. They were simply unable to anticipate the large-scale attack and losses they'd face by the Israeli air force's attack. But what may have harmed their war effort even more was a tidbit I only recently noticed in some other reading: the best Syrian brigades (the 5th and 70th) were kept in their barracks at Homs and Damascus. Hafiz Assad, then the war minister, begged to send them to the front, but was stopped by the head of government (Salah Jadid). Apparently, Jadid physically assaulted Assad, before telling him that saving a few square miles of territory was not worth risking the fall of the regime. Jadid was fearful that because his regime was not popular with much of the population, sending them off would open up the possibility of a coup or uprising at home. This is my speculation, but it's particularly possible that Jadid feared that if they were destroyed, he would have no bulwark against any coup. And this would be particularly poignant, since Jadid used those two brigades to displace Hafiz Assad back in 1966, when Assad's brigades were away from Damascus (or had been infiltrated by Jadid's men). So this also may have contributed to some of the Syrian losses and seeming inaction, though not by much.

Can anyone recommend some good Arab-Israeli conflict books?