As an aside, I love that I'm gonna be able to answer this question.

As another aside, the question hasn't really been answered before, which makes me more excited!

It's important for me to give you the basic facts, just in case you (reader!) are following along and don't know about the incident! I will do background, then go into the Israeli side, then the American side, then go into possible motives and possible explanations of inconsistencies. To skip to any of the sections, ctrl+f any of the following (this is the order they're in):

"Background"
"The Israeli Story"
"The American Story"
"Motives and Explanations"

Oh, and for anyone looking for an online document format of this (Reddit formatting might not be the best for you!), here it is: http://www.scribd.com/doc/221490139/The-USS-Liberty-Not-a-Conspiracy-After-All

Enjoy!

Background

The Six Day War between Israel and Egypt had begun. Fighting had also begun in the West Bank, with Jordan. Fighting would be slated to begin the next day with Syria, and Syrian and Israeli forces were prepared for it. Now, I'm not going to comment on the war's beginnings, or the justifications or lack thereof, because that's a post in and of itself. This is the backdrop, however.

The Israeli Story

Now, this is drawn from the Yerushalmi Report.

On Thursday 8th June, 1967, at approximately 1400 hours, aircraft of the Israel Air Force attacked a vessel situated about 20 miles north-west of El-Arish and some 14 miles off the shore of Bardawil. About half an hour later torpedo boats of the Israel Navy attacked the same vessel and hit it with a torpedo. Soon, during the attack by the torpedo boats, it became clear that the vessel, thought to be an enemy ship, was a vessel named "Liberty," of the United States Navy. The attack was immediately broken off, but, most regrettably, only after, as transpired, loss of life and material damage had been caused.

Now, some more background to help explain how this happened, drawing from the report.

It was the 4th day of the war, and the towns of Gaza and El-Arish were already in Israeli hands. Naval engagements were taking place along the coast of Israel.

Two different sources radioed Navy H.Q in Israel, saying that El-Arish was being bombarded from the sea, from Southern Command.

At 1205, three torpedo boats were given orders to proceed to El-Arish. Shelling reports continued to reach Navy HQ, and pressure mounted to react. Claims came in that "El-Arish has been shelled for hours and the Navy is not reacting!", to explain some of the frustration.

Air Force HQ and Navy HQ were coordinating, and agreed that when the torpedo boats located the target, Air Force would send aircraft.

At about 1341 hours, a target was located 20 miles north off the coast of El-Arish. The division reported that the target was moving 30 knots westward towards Port Said.

At this point, a map may come in handy. This one marks El-Arish (roughly, obviously). Port Said is right next to Port Fuad. They are roughly 167 km apart judging by driving directions Google provides. At the rate they were going, it would likely take about 3 and a half hours to reach Port Said. 30 knots is roughly 55km/h.

A few minutes later, the Division Commander reported that the target, now 17 miles from him, was moving at 28 knots. He reported he could not overtake it, and requested aircraft. He lastly said that it had changed direction.

Aircraft flew over it, looking for a flag. They said they found none. They noted that it had two guns, and that it was gray (which is a warship color).

Smoke and fire were seen after the first strafing run by the aircraft, which was done with cannon, guns, and bombs. The aircraft was ordered then to leave the target, to allow the torpedo boats to engage in attack since they drew near during this attack. A low-flying aircraft saw CPR-5 on the hull. An order was then transmitted to the torpedo division not to attack, as it might have been misidentified.

The Division Commander was ordered to get closer to the ship, and identify it. The order was carried out, and the Commander reported that the ship appeared to be a merchant or supply vessel. The Division Commander also signaled the ship and requested its identification, but the latter replied with a signal meaning "identify yourself first."

Meanwhile, the Division Commander was looking through a book used to identify Arab naval forces, and concluded that this was an Egyptian supply ship named "El-Kasir". Another torpedo boat commander radioed saying that he had identified the ship as "El-Kasir". Taking this as fact, Division Commander authorized the torpedo boats to attack at 1436.

A torpedo was fired, and hit. Only later would the torpedo ships approaching from the other side see the markings "CTR-5", at which time Division Commander called off the attack.

The Israelis insisted there were no American flags displayed on the ship, and only an assisting helicopter after the attack saw a small American flag flying on the ship at all, when it was identified as an audio-surveillance ship of the US Navy.

The Israelis then saw the ship leave the scene of its own accord, without requiring the assistance offered. However, it is fair to assume, all things considered, that there was considerable damage to the vessel and material and life lost.

The next comment continues the Israeli side.

I cannot find any hard evidence showing that it was truly an accident, but I have to stay with my side.

To add to /u/BeerMania, no one documents that they're doing accidentally. Then it wouldn't be an accident. Besides the fact that the shift-changed is documented to have caused confusion, there are a few important notes.

First, Israel was undergoing a very quick advance. As a consequence, confusion was likely to be rampant. The IAF bombed an Israeli armored column around the same time, showing the confusing nature of things. And that was on the Jordanian front, which was much smaller than the Egyptian one (the only citation I can find is a news article--I'm trying to find something more solid).

Second, there's no realistic cause for it to happen. The usual hypothesis is that it was an attack to blame on Egypt and get the US to enter. But the US was already supporting Israel, and wasn't going to get involved because that would've meant Soviet intervention directly. Besides, if it was a "false flag", surely the planes and ships would've actually sunk it, and been told to attack no matter the identity. They weren't. Why would they have the attack stop when the identity was discovered? Why not conceal that it was Israeli, which was clear from the outset? The way things progressed, it's pretty clear it wasn't centrally planned. If it was a false flag, it was a poorly executed one, and half-assing something like that doesn't make much sense.

About the 12th paragraph then just keep reading. The changing of the duty guards. Mostly about the explosion off the coast of the Sinai. They thought the coast was being shelled. & there is just so much information in there that I can't give you a tl:dr.

http://www.jewishvirtuallibrary.org/jsource/History/liberty1.html

The IDF launched lightning air and ground strikes against Egypt, quickly gaining the initiative, and repulsed attacks from Syria and Jordan. Yet the Israelis remained highly concerned about threats to their coastline, along which most of the country's major industrial and population centers were situated. The Egyptian navy outnumbered Israel's by more than five to one in warships and, in a crisis, could call on the support of some seventy Soviet vessels in the vicinity.14 The failure of the Israeli navy's attacks on Egyptian and Syrian ports early in the war did little to assuage Israel's fears. Consequently, the IDF Chief of Staff, Gen. Yitzhak Rabin, informed the U.S. Naval Attaché in Tel Aviv, Cmdr. Ernest Carl Castle, that Israel would defend its coast with every means at its disposal. Unidentified vessels would be sunk, Rabin advised; the United States should either acknowledge its ships in the area or remove them.15 Nonetheless, the Americans provided Israel with no information on the Liberty. The United States had also rejected Israel's request for a formal naval liaison. On May 31, Avraham Harman, Israel's ambassador to Washington, had warned Under Secretary of State Eugene V. Rostow that "if war breaks out, we would have no telephone number to call, no code for plane recognition, and no way to get in touch with the U.S. Sixth Fleet