1) The use of Eagle in ancient Rome seem to symbolize imperium, power or authority and military power. Roman legions had carried the aquila, or eagle, as signa militaria. Under the later emperors the eagle was carried, as it had been for many centuries, with the legion, a legion being on that account sometimes called aquila. Each cohort had for its own ensign the serpent or dragon, which was woven on a square piece of cloth textilis anguis, elevated on a gilt staff, to which a cross-bar was adapted for the purpose, and carried by the draconarius.
Eagle feathers are often used in various ceremonies and are used to honor noteworthy achievements and qualities such as exceptional leadership and bravery.
2) Constantinople was the successor of Rome, and the Byzantines continued the use of the old imperial "single-headed" eagle motif as a symbol of their empire and If it's a double headed eagle therefor it is a common symbol in heraldry and vexillology. It is most commonly associated with the Holy Roman Empire, the Byzantine Empire and similar to the Vijayanagara Empire's gandaberunda. Eagles have been used by many nations as a national symbol.
3) Snake could be a symbol of healing, a sign of God's power, wisdom and knowledge.
And when these two symbols were put together, its kind of resemblance of a strong ruling power.