It allows the states to manage all the voting.
Basically, a state – we’ll pick Vermont – has 3 electoral votes, because its population gives it 1 seat in the House of Representatives and its status as a state gives it 2 seats in the Senate. Exactly how Vermont chooses to apportion those 3 votes is up to the government of Vermont. It’s theoretically possible that Vermont could pass a state law saying that Vermont’s 3 electoral votes go to the candidate of the same party as the sitting governor.
In the early days, when this system was devised, it was done to ensure that the several states still had considerable power. The framers of the Constitution saw that centralized power had good aspects and bad ones from their point of view, and tried to minimize the centralized power in such a way that it enabled the good aspects and put checks on the bad ones. Keeping the states involved as states in the electoral college was one of the ways of accomplishing this.
More recently, one of the major benefits is that it allows the elections to be certified and valid. Suppose that in the 2008 Presidential election, Vermont buys voting machines from a totally incompetent manufacturer, and this isn’t revealed until the night of the election, when Vermont, with a population under a million, reports that 120 million people have voted for the Republican and 135 million people have voted for the Democrat—and worse, as there’s no paper trail, they can’t establish how many people actually voted or who they voted for. If the margin of victory in the national election is under 200,000 votes otherwise, the whole thing would be a disaster under a simple majority system—there’s no way of knowing whether Vermont would sway the election or not. But under the electoral college system, it’s possible to say, well, the Republican has 315 electoral votes, the Democrat has 217, even if the state went strongly for the Democrat, the Republican would still win nationally.
On another level, it uses the two-party system adversarially to minimize voting fraud. Once a party gets to 51% of the vote—enough to cast the whole state’s electoral college votes for that candidate—there’s no point in further cheating. Otherwise, there would be a significant temptation for a party that strongly controls an area (Democrats in Massachusetts, for instance) to conduct widespread vote fraud and inflate the number of votes for their candidate significantly, “cancelling out” votes for their opponents from other areas.
So there are several benefits to doing it this way. Whether they outweigh the benefits of “one man, one vote” approaches is a different question.