It's counterintuitive, but spring weather is usually pretty overcast and grey, which is what this comes from I think. You can get more of the lore on this page: https://groundhog-day.com/history-of-groundhog-day

Here's a quote:

> Spring weather can be pretty miserable — oftentimes it’s grey and rainy and wet — whereas the middle of winter has plenty of bright, clear days where it is insensibly cold outside. Essentially, the Candlemas prediction assumes that overcast weather is a harbinger of spring, whereas a clear day means you’re still in the thick of winter.