Why professors spend most of their time to write proposals than doing research?
I'm a bit puzzled by the word "hidden," since I don't think the answer is at all hidden -- in fact, I often discuss this with my students. Why spend lots of time writing proposals?

(1) It's expensive to run a lab! At least in the U.S., a graduate student typically costs ~$50k/year, which includes tuition and overhead costs that the student doesn't see. Postdocs are ~$100k/year. Equipment and supplies are not cheap. The sum of all this is a lot of money.

(2) For a variety of reasons, it's hard to get money. The success rates for proposals at NIH or NSF are around 10-20%, meaning that most proposals don't get funded. Certainly more than 20% of proposals are very good -- at least based on my experience on grant review panels. Writing a proposal that's at least very good takes a lot of work. Writing N such proposals so that enough get funded to keep one's lab afloat takes an even larger amount of work.