The eagle is in reference to Polish legend about its founding.

The legend is about three brother rulers Lech, Czech and Rus, who set out to find new homelands for their people. Lech is the leader of the Polish, Czech of the Czech and Rus of the Russians. Lech is said to venture West and find a suitable place when he comes upon a white eagle's nest upon sunset. The rays of light passed through some of the feathers, illuminating them in a gold like shine, with the rest of the eagle pure white against a red backdrop.

Edit: I didn't address why the HRE uses the eagle, sorry. They based it on the Roman eagle standard used in legions.

The main HRE imperial coat though has a double headed eagle which is worth mentioning.

Also, the Silesian Piast dynasty (fragmentation of the original ruling dynasty of Poland) used the same stylized eagle except gold on a blue background, and then used a black eagle on yellow backdrop when they came into Bohemian sphere of influence.