In research proposals, why do investigators, who usually are permanent faculty members, need to claim a certain fraction of their full-time salaries, from the proposed budget? Are they not paid in full by the university?
There are several reasons in the US. Most faculty are paid for 9 months of the year not 12. They may choose to spread out their pay into 12 payments a year, but their base salary is for 9 months. Grant funding is one way to supplement their pay up to a full 12 months. Unfortunately, the National Science Foundation, which funds a lot of academic research in the US, only allows a maximum of 2 months to come from NSF, so professors must diversify somehow as well.
Additionally, some faculty may choose to buy out some of their required teaching. Many departments will allow you to teach less if you can bring in even more of your salary. Sometimes this is done with grant funding, but I think more often it is done using endowed chairs.
This varies by the industry, as well. Many NIH funded researchers in the biological and health sciences are effectively full time researchers, and their salaries are not funded by their institutions directly (rather, are often 'paid' in laboratory space). NIH grants pay their full salary (up to the NIH Salary Cap, linked to here for 2015).
Institutions might give these professors a constant salary from the institution, with the expectation that most/all of that salary is reimbursed through grant work. (Ignoring tenure, for the moment; of course a tenured professor may receive a salary even if they are unable to find sufficient grant work for an extended period of time.)
In the UK at least, it is because the funder is contracting with the university for their employee's time. This PowerPoint from Birkbeck, for instance goes into costs etc, plus the requirement for institutional sign-offs. The investigator will continue to be paid by their university as before (normally) but the University will be compensated for the relevant time.