Origins
The next time you feel eager about something, know that you are — etymologically — being sharp. The word comes from Old French aigre meaning 'sour, acid, fierce', from Latin ācer meaning 'sharp' or 'keen'.
In 13th-century English, eager meant 'fierce'. A hawk was eager — sharp-eyed, sharp-clawed, ready to strike. An eager wind was a biting one. The warmth of modern eagerness was not yet present; it was all edge and intensity.
The Proto-Indo-European root *h₂eḱ- meant 'sharp' or 'pointed', and it produced an extraordinary family. Through Latin ācer we get acid (sharp-tasting), acrid (sharp-smelling), acute (sharp-angled or sharp-minded), acme (the sharp peak), and acumen (sharp judgement).
Later History
Perhaps most surprisingly, vinegar belongs to this family too. French vin aigre means literally 'sharp wine' — wine that has turned acid. The aigre in vinegar is the same word as eager, just centuries apart.
The shift from 'fierce' to 'enthusiastic' happened gradually during the 16th century. The sharpness remained as metaphor: an eager person cuts through hesitation.