Historical associations with power and strength, mostly.

First off, the Eagle was an important emblem for the Romans. You can read more about why that happened in this question with answer by /u/The_NSA_are_nice, but tldr: there were a couple of animal symbols in the beginning, but the eagle gradually becaome prominent as the symbol of empire.

The idea that eagles represent power and strength has continued since then. This association became somewhat twofold- the Romans considered the eagle to be powerful and strong, and a lot of subsequent civilizations considered the Romans to be powerful and strong. The link between Romans and eagle is also pretty well known. Most people will know of the Eagle standards carried by legions, or have seen the SPQR design that features an eagle prominently. So using the eagle is also an easy way to associate yourself with the Romans, and man, people really want to associate themselves with the Romans.

It's fame in America started with its inclusion in the national seal, the original design of which you can see here. The role of the eagle again grew quickly over time, also largely for the connotations of power and strength.