When temperatures drop and frost hits, only the heartiest of veggies are equipped to survive. And for many of them, the adaptation that keeps them from dying in the cold also makes them sweet and delicious.
In the above video, UCLA biochemist Liz Roth-Johnson explains how this works for the crunchy carrot: When it gets cold out, carrots (and parsnips) convert some of their starch stores into sugar. They do this to keep the water in their cells from freezing, and it works in the same way that putting salt on a road keeps it from freezing over. When a foreign substance mixes with cold water, it makes it harder for enough water molecules to reach the surface and freeze there -- so the freezing point gets lower. The cells inside a carrot might have icy-cold water, but that water won't turn into ice.