Isn't it "writing things down" that helps you think? Does it have to be "hand writing" in particular?

Writing things down is handwriting. You perhaps mean “typing things out” as in using a keyboard. You get some benefits from typing things out but you don’t get many others, as handwriting requires you literally think about each letter and word as it is constructed and which actions you need to take to construct the letters and words. If I type, I type it and forget about it by design moving on to the next word not worrying at all about it. Handwriting essentially trains you to see, think, and understand longterm chains of thought; essentially deep focus on what you are doing while thinking about what is coming next. Typing only produces shortterm. Shortterm thought pairs well with brain dumping, such as on Twitter, which ends up being pretty valueless in the longterm.