The shield that you see in the center of a national coat of arms is called an escutcheon, and it shows the heraldic device that represents that country. In the full coat of arms, the escutcheon is usually surrounded by heraldic animals, called supporters. For example, in the coat of arms of the United Kingdom, you have a golden lion and a silver unicorn on the left and right of the escutcheon, both of them in the rampant posture, basically holding the escutcheon up. In the case of Austria, USA, Egypt, etc. the supporter is an eagle in the displayed posture, and the supporter "holds" the escutheon placed on the chest, in the center of the eagle.

The reason for the eagle displayed (and it is an eagle in all these cases) is in part that there is no reason. Heraldry does not always have a meaning, and the use of specific animals and devices tends to be more grounded in tradition than in a specific meaning. This said, the eagle was a symbol of the Roman Empire, and it was used in European heraldry from the central middle ages, where a black eagle displayed was the heraldic symbol of the holy Roman Empire. The eagle as a supporter in the coat of arms of several European countries can be variants traced back to the eagle of the HRE, whether they were countries that were at one point part of the HRE or with royal lines that were somewhat associated with it.

The eagle in the coats of arms of Egypt, UAE, Syria, and other arab countries is more a symbol of pan-arab nationalism. It is called the Saladin eagle, even though I am not aware of any evidence that actually traces it back to Saladin himself.

If you want to know more about heraldry, the 100+ year old Fox-Davies, A complete guide to heraldry is actually a decent introduction to the topic.