Because they’re sweet. Next to sugar beets, carrots contain more sugar than any other vegetable. Being root crops, they have a longer seasonal availability than fruit, and once harvested can be kept longer. No wonder then that carrots have played a starring role in a number of sweet preparations over the centuries, at least when purer sweets like sugar or honey weren’t available.
Looking back through the annals of food history, one can see that sweet carrot stews were popular in the Old World prior to the Colonial era, before cane sugar from the Caribbean became widely available. Carrot puddings and/or cakes were intermittently popular in Britain and America during the 1700’s, a time when on-again off-again conflicts interrupted trade and/or caused the price of sugar to fluctuate wildly.
Starting in about the early 1800’s, carrot cake and pudding recipes disappear entirely in America, for reasons that aren’t hard to fathom. Sugar was abundant in that period and continued to be so right up through the first and second world wars when — and you can see where this is going — sugar became scarce once again. It’s at this time that carrot cakes as we know them first appeared on the culinary scene.
So austerity and carrot cake are historically linked. Or at least they were until the 1960’s, when the countercultural movement’s voluntary austerity (and health consciousness) brought it back again. Is it here to stay? That remains to be seen.