With eager, think "ready," "excited," "can't wait." An eager beaver is someone who, like the hard working beaver, is up at dawn and ready to get to work.
Eager derives from old words meaning "sharp, pungent or keen," and eager carries that sense of sharpness still. In fact, being overeager is a criticism; someone overeager is likely to make a mistake because they are moving too quickly or not thinking things through before they act.
Primary Meanings of eager
having or showing keen interest or intense desire or impatient expectancy
a high wave (often dangerous) caused by tidal flow (as by colliding tidal currents or in a narrow estuary)
adj having or showing keen interest or intense desire or impatient expectancy
having or showing great eagerness or enthusiasm
(usually followed by `to') full of eagerness
having or showing great excitement and interest