Logical reason why writing ideas down is important

It's arithmatic, actually.

So, you're thinking of ideas for a project, a game, a story, etc. You're thinking all day, but at a certain point, it becomes hard to build off your ideas. 

You're wasting mental storage space- and energy - on remembering the ideas you've already thought. It's overloading your short-term memory. It doesn't always go to your long-term memory, that's why we can recall plenty of times where we can't recall a great idea we've had.

So, as soon as you can, write those first ideas down, clear out the freezer, fridge, safety-deposit box, washer, dryer, whatever. 

I would say, 75% of the time, at the very least, you'll come up with more ideas while you're writing down the ones you have. Your brain automatically starts making connections to more ideas. It won't do this, most often, if it risks losing previous knowledge.

So, yeah. It's math, baby!