There are probably 2 reasons for this:

1. There’s probably a lot more motor control going on than you would expect when you need to think (writing, fidgeting, etc.).

2. Your brain wants sugars, so when you run out of immediately-available glycogen to break down, you will want to eat more in order to keep thinking. Breaking down fats wont supply energy fast enough (in the short term) to keep complex thought running continuously.