The modern sense was around in Shakespeare's time - it had been in English since c1400 - and he (or someone) used it in King Edward III when Salisbury talks of being "eager of revenge" (V.i.115). Today only people can be keen, not things or conditions - but in Early Modern English, anything could be eager. When Hamlet talks about the "eager air" (Hamlet, I.iv.2), he means it is biting. When George talks to Richard of vexing dead Clifford with "eager words" (Henry VI Part 3, II.vi.68), he means his words are cutting. In the Quarto texts of Hamlet, the Ghost tells Hamlet of a poison "like eager droppings into milk", meaning "sour". And King David talks of beating the English "with eager rods" (King Edward III, I.ii.25), meaning "fierce". The word had a much more intense set of meanings than it has today, and this intensity is found when it is used of people too. An "eager cry" (Richard II, V.iii.74) is more than just mildly excited - it is impetuously impassioned.
David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin. A subscription website is available at www.shakespeareswords.com