"If your stock is accessible to more people, there's higher demand" - this the answer 90% of the time, but I don't believe there has ever been a study (and I've read three or four) that suggests there is any evidence this is true.
To make what I consider to be a not unreasonable comparison, "The standard gold bar held as gold reserves by central banks and traded among bullion dealers is the 400-troy-ounce (12.4 kg or 438.9 ounces) Good Delivery gold bar." [1] - that gold bar currently sells for $41,260 [2] * 12.4 or $511,624/bar. There does not seem to be any difference, per ounce, in price between a one ounce nugget and a 438.9 ounce unit (indeed, there seems to be some transaction costs that imply the larger bar is less expensive per ounce)
For a mature company that is throwing off lots of profit and cash , there are a lot of investors who are analyzing the Discounted Cash Flow of that company, and paying for the stock based on that number. Apple, for better or worse, has entered that realm. Therefore, if they trade significantly below their cash-flow value, those investors will buy up the stock, if they trade significantly above it, those investors will sell the stock (or short it if things get too far out of whack) - It doesn't matter if the stock is selling for $100, $1000, or $10,000/share.
But, the people who run Apple have access to the best advice that money can buy, and they have made this decision - what I'm interested in, ideally from someone who has worked behind the scenes in this kind of really-big-company stock split, is the real reason why this is done. I have a theory it's a mechanism to reward bankers with a lot of business with useless make-work, in return for getting positive ratings moving forward - but I'm open to other theories.
[Edit - another Theory is that Tim Cook and/or his CFO Peter Oppenheimer are just fed up with being asked about a stock split, and given that the stock split is an absolutely meaningless event, that has no negative/positive impact on the company value, but will eliminate the ongoing nagging of people about it, just decided to eliminate that annoyance. My counter argument to that would be is by giving people something meaningless to complain or ask about, you distract them from other things that you would prefer they not spend so much time on - so it's good to leave a few of these around as decoys.]
[1] http://en.wikipedia.org/wiki/Gold_bar
[2] http://www.bullionvault.com/