I believe the date you reference uses the Byzantine calendar to reckon its dates. At this point in history (AD 1448) the Hagia Sofia was used by the Greek Orthodox church, and this calendar was the official calendar of the Byzantine Empire (which controlled Constantinople at the time in question).

They based their reckoning of year on an Anno Mundi epoch (means they dated their years from the date they believed the world was created), so 6956 would have been 6,956 years after the creation of the world, and September 1, 5509 BC would have been the date of the creation of the world. Now, for dates that seem a year off, remember that the Byzantine calendar begins on September 1, so there is a large 9 month lag, and so that could change dates by a year in our system. For example, a date in September or later 1448 would be the beginning of Byzantine year 6957, but in Augustor earlier 1448 that would be the end of the previous year, so 6956. That's why some of the dates seem a year off to you.

Why did they pick September 1 5509 BC as the date of creation? They derived the date from the Greek version of the Old Testament, the Septuagint, which was used by early Christian scholars (credited to Theophilus). They used the Biblical chronologies to reconstruct a chronology back to creation. You can read Theophilus' work in the Apology to Autolycus, book III chapters 24-27. Bishop James Ussher did a similar thing much later, revising the date of creation back to 4004 BC..