If you do something eagerly, you do it with much excitement and enthusiasm. When it's your turn to hit the piñata at a party, you might grab the bat eagerly because you can't wait to take a swing.
If you skip breakfast and lunch, you'll eat dinner eagerly, and if you collect stamps, you might eagerly examine every stamped envelope you come across, in case you can add it to your collection. Anything done with an eager attitude is done eagerly. Interestingly, eager has changed in meaning over the years — in the 13th century it meant "fierce or angry," from the Latin acrem, "pointed, piercing, or zealous."