With the now-cold war against the Spartans being in its 16th year many Athenians were looking for a way to assert themselves. Alcibiades gave them the excuse they wanted by arguing that Athens should come to the aid of the Ionian Sicilians of Egesta and protect them against the Doric Sicilians, most notably Syracuse.

The city of Syracuse was targeted because it was very wealthy and, ultimately, a potential threat. Alcibiades and his supporters argued that their potential rise as a naval power could eventually threaten that of Athens, saying "lest if others be not subject to us, we fall in danger of being subjected unto them" (Thuc. 6.18). For him, and many other young Athenians, the invasion of Syracuse was a chance to make their mark on history and achieve glory for themselves and Athens. According to Plutach, Alcibiades also considered Sicily a stepping stone to the creation of a larger empire, with his sights set on eventual conquest of Carthage and Libya: "Many were they who sat in the palaestras and lounging-places mapping out in the sand the shape of Sicily and the position of Libya and Carthage" (Plut. Alc. 17.3).

There was opposition to the plan, as Nicias preached caution, saying, to quote Thucydides' account of his speech, "I say that going thither you leave many enemies here behind you, and more you endeavour to draw hither." Nicias warned that with the uneasy peace in its 6th year, an attack on an ally of Sparta might reignite the war, and that should they invest such a large force in the expedition they risk being weakened at home (Thuc. 6.10). He also attempted to dissuade the assembly by pointing out the great many men, ships and supplies that must be committed to ensure success, but instead -- hilariously -- only served to make the Athenians more excited about the expedition (Thuc. 6.24).

Ironically it was Nicias who was elected general for the expedition alongside Alcibiades to temper his youth with the elder's "prudent forethought" (Plut. Alc. 18.1). Alcibiades himself wouldn't see the invasion through, however, as he was called back to Athens early to be held trial for the vandalizing of the Herms (statues of Hermes).

Without simplifying things too much, the Sicilian Expedition was seen by many Athenians as a chance to recapture their greatness and reassert themselves as the preeminent Mediterranean power. This was despite many older Athenians wanting the hazy peace with Sparta to remain. A lazy comparison could be made between the Peloponnesian war and the Cold War of the last century. Both were fought in proxy states, and both featured few direct confrontations. Sicily was another one of these proxy states, and one that cost Athens dearly.

Thucydides covers this in the Sixth Book of the History of the Peloponnesian War. Here's a link: http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0247%3Abook%3D6%3Achapter%3D8