Why do researchers in areas requiring minimal resources apply for grants? I heard that many researchers spend a lot of their time (maybe even a half) writing the grants. I understand the researchers that need to purchase some equipment or fund their PhD students, although does not the department fund PhD students anyway?

But what about pure mathematicians and researchers who need a pencil and paper, a personal computer. Do these apply for grants, what for?
I am a theoretical computer scientists in a top-10 US computer science department.  To first order approximation, the only reason I apply for grants is to fund PhD students.


My research does not require PhD students, but it's part of my job as a university professor to advise them, I enjoy working with them, and I am considerably more productive when I work with them.
Students in my department that are not funded by research grants or fellowships are funded instead by teaching assistantships.  Being a TA requires a significant time investment, decreasing the time that these students can devote to research.  Being a TA can be incredibly valuable experience, especially for prospective academics, but the first-order criterion for judging the success of PhDs in computer science is the quality of their research output.
My department competes with peer departments — and I compete with researchers in those departments — to attract strong PhD students.  If we/I want to attract strong students, we have to make credible promises of future funding to do research, because that's what our peer departments do.  Prospective PhD students understand the previous point.
I work in a public university in a state with rather significant budget problems.  TAs are paid from state money, which is limited.  Thus, my department can only support a limited number of TAs.  So if I want the department to admit more than a small number of students into my research area, I have to demonstrate that we can fund a majority of those students through grants.
Students need funding to travel to conferences to present their research results, because they need to build a reputation in the research community.  I could pay for my own travel out of pocket if I really had to, but most PhD students don't have that luxury.


There are a few second-order concerns as well.


The day-to-day functioning of my department requires a steady stream of incoming research grants.  Grant overhead pays for a lot of basic infrastructure in my department, including the salaries of all the staff our business office, half of the salaries of most other administrative staff, support for non-instructional non-research computing resources like our graduate admissions database.  Grant overhead is also the source of startup packages for new faculty (for which we are again competing with our peers).
The only equipment I need is "pencil and paper"—or more accurately, a laptop with a stable LaTeX distribution and a drawing program, and basic internet access.  NSF no longer allows research funds to be used for general-purpose computing equipment, so I can't include that in my grant budgets anyway.  But my university returns a small fraction of overhead directly to PIs, and I can use that to buy new laptops, to pay for additional travel (by me, my students, or visitors), or even to help fund RAships.
Tenure and promotion decisions, and to a smaller extent raises, do depend in part on professors' proven ability to attract funding, especially in a department (and college) like mine where most faculty are extremely successful.
Oh right, I almost forgot.  I can pay myself an extra month or two of salary over the summer from my grants.  The university pays me for only 9 months of each year (although that pay is spread over 12 months), so each month of summer salary is actually more than 10% of my annual pay.