I had exactly the same problem and was pointed to the system requirements, which Apple telephone support said were on the box - not as far as I can see. I then looked at the website, which extols the virtues of the iPhone 5 but you have to really search for the technical specifications to find the above information.
Having recently bought an iPad 3 which works perfectly, I was astonished that the new iPhone did not. So if you have a perfectly good Apple computer which is over 6 years old Apple appears not to want to know you. Interestingly my ancient PC running WindowsXP would appear to work fine, so the answer appears to be that my next computer purchase will be a PC not a Mac - so much for 25 years of undying loyalty - and probably an Android phone after I sell the iPhone on eBay if Apple don't sort this out soon.
A partial workaround to transfer Calendars, Contacs, Apps and settings etc, assuming you are upgrading as I have, is to backup your old iPhone to iCloud and then reset the iPhone 5 and restore from the iCloud backup. However, I still cannot work out how to get my music across without paying for an iTunes Match subscription, which I find annoying. If I had a PC, there appear to be a number of solutions to transfer music files but I have yet to find a Mac one that works. Has anybody out there found a solution?
I think my long Apple honeymoon will soon be over if they don't get this sorted. Pity really but if they want to behave like Microsoft why bother to buy Apple in future.