Why is YouTube so slow on rainy days?

It is possible that you are using satellite-based internet. For such an instance, the internet would slow down due to high moisture content in the air obstructing the satellite and thus interfering with the signal. Much like how DirecTV customers often lose service when a huge storm starts up.

If you’re not using satellite-based internet, then I’m quite puzzled. I’ve never had problems with my copper wire internet during a storm.

It's likely that there are larger quantities of customers using greater volumes of bandwidth in your area to your local cabinet or centralisation point (depending on where you live) due to a lack of desire to venture outdoors into the less pleasant weather. Unfortunately, bandwidth will always remain a hindrance in the foreseeable future.