Why did Xerxes need a canal across Mount Athos peninsula?

Alex's answer is correct, more or less. I will add the following:

The problem with Athos is that it is very high, a sheer, conical rock and the waters around Athos are littered with sharp, craggy rocks. Therefore, when sailing around it you have to give it wide berth and if a storm or strong winds arise you are screwed. In that area the North wind is the dangerous wind.

The other two peninsulas, Sithonia and Pallene, are much safer because they are shielded from the wind by Athos, and both have multiple harbors in which ships can hide if a storm occurs.

The main error in Alex's answer is that describes the need to land frequently as the key factor. This is not exactly right. Athos can be rounded in one or two days by even a slow ship without needing to put in. The danger is purely that strong winds could occur while you are rounding the cape. A related problem is that strong winds can prevent a passage and delay ships. With the canal there is no delay and no chance of being caught out by a surprise gale.

One thing to remember is that war has a big logistical aspect to it. During certain times of the years there can be strong winds that go on for days in the North Aegaean. When you are a military commander one thing you do not want to hear is, "Oh, sorry, sir, we can't send supply ships for two weeks because we have to wait for the winds to die down first."

Careful examination of the Google Earth photos led me to the following conjecture. The difference between the Mount Athos peninsula, and the other two is that there are no beaches on the Mount Athos peninsula. The mountain rises steeply from the water. This means that one has no landing for about 70 miles if one tries to sail round it. The other two peninsulae seem to have convenient beaches, as I can judge from Google Earth photos.

The problem with the ships of that time was that they had to land frequently. They did not sail at night, and did not sail in bad weather, and were enormously overcrowded, so that they could not carry sufficient food or water supplies. Therefore, rounding Mount Athos peninsula with no landing places could be a challenge at that time.

Despite all these enormous preparations, in his invasion Xerxes lost a substantial part of the fleet to adverse weather conditions. This could be avoided by starting operations earlier in summer. It is very strange that Xerxes advisers did not understand this: there were experienced sailors among them!

The waters around Mount Athos were known to be dangerous.^[Note] It was here that an earlier Persian fleet met its demise during the invasion of 492 BC. According to Herodotus in his Histories (VI. 45):

Crossing over from Thasos [the Persians] travelled near the land as far as Acanthus, and putting out from there they tried to round Athos. But a great and irresistible north wind fell upon them as they sailed past and dealt very roughly with them, driving many of their ships upon Athos. It is said that about three hundred ships were lost, and more than twenty thousand men.

When Xerxes ordered the construction of the Athos canal during the Second Invasion, it was explicitly described as an effort to avoid the the same danger (VII. 23).

Since those who had earlier attempted to sail around Athos had suffered shipwreck, for about three years preparations had been underway there. Triremes were anchored off Elaeus in the Chersonese; with these for their headquarters, all sorts of men in the army were compelled by whippings to dig a canal.

From a practical perspective, then, the rationale for the canal was safety. On the other hand, Herodotus also attributed the effort to Xerxes' pride (VII. 25):

As far as I can judge by conjecture, Xerxes gave the command for this digging out of pride, wishing to display his power and leave a memorial.

Either way, the canal was not dug to reduce the travel distance required. That said, Xerxes apparently failed to understand the root causes of the danger. His fleet was mauled in essentially a repeat of the same mistake further down the line at Sepias (VII. 189).

_{Note: "High seas and dangerous currents are found off Mt. Athos ... but it was strong winds that initially caused the disaster. The wind in question was Boreas, a fierce northerly produced by the same juxtaposition of cold dry high pressure air over the continental interior and warmer moist low pressure air over the sea which gives rise to the violent Bora in the Adriatic ... the Persian fleet was always in danger of being blown directly onto its northeastern shore." - Morton, Jamie. The Role of the Physical Environment in Ancient Greek Seafaring. Brill, 2001.}