The eagle or aquila as it was called in Latin, was actually just one of five animals used to represent Rome as part of their standards. Pliny the Elder mentions five animals being used to represent Rome, the eagle, wolf, boar, minotaur, and the horse. At Gaius Marius' second consulship those present decided that the eagle was to be chosen as the one standard to represent Rome. So while we think of the eagle as their standard, that was more of an empire thing rather than a republic thing. On to your actual question on why it was chosen, was because of the qualities that the eagle was said to have possessed/represented. The eagle is one of the most powerful birds in the world and is at the top of the food chain in many of its habitats therefore it represents dominance, power, courage, bravery, etc. You can get the idea that the Romans wanted to be seen as this awesome army that would never back down from a fight, would always emerge victorious, and would pursue its enemies till the end. In many ways they picked the eagle because they felt that it represented what their culture and military in particular valued. The individual legions became so protective over their standards that a position was designed for the sole purpose of watching over the aquila and carrying it into battle. The aquilifers were under the centurions of a legion but were above the Roman equivalent of a lieutenant (optiones) and received many perks that regular soldiers weren't entitled to such as better meals as well as much higher pay. The aquilas, if lost were to be retrieved at all costs. If an aquila is lost it was equivalent to losing the spirit of a legion and was usually devastating to a legion to see their standard in the hands of an enemy. They would attempt to retrieve a lost aquila at all costs, oftentimes coming back years later such as at Teutoberg Forest to retrieve their lost aquilas.

TL;DR Eagle represents power, courage, and bravery all of which were idolized by the Romans so they adopted it as their standard

Did the symbol of the eagle continue on to the Holy Roman Empire and consequently- to Germany?