The phrase means a long time and it originated in the early 20th century, apparently as a pun on the long ears of a donkey. In fact, the first published reference in the Oxford English Dictionary uses the phrase “donkey’s ears.” Here’s the citation, from The Vermillion Box, a 1916 novel by E. V. Lucas: “Now for my first bath for what the men call ‘Donkey’s ears,’ meaning years and years.”