Other answers here are useful but incomplete and I believe actually miss the main thrust of what the original quote is referring to.

Despite wide support to pursue research as an academic, there are limitations on the amount of time most academics can commit to their research. A primary way to expand the proportion of your time that you can devote to research, as well as to resource projects that could not otherwise be undertaken at all, is to apply for research grants, primarily research grants external to your employing institution.

However, the vast majority of such grants are awarded for specific research projects or fairly defined research programs. Funding agencies virtually all require these to be defined to a fairly specific degree of detail in the competitive application process. When funding is awarded, it is an expectation of the funding agency that the research that is undertaken will be the work that is funded, and there are typically also reporting mechanisms that can be used to monitor this. While the specifics will vary between disciplines, such an approach by definition leads to relatively 'monolithic' research projects, where the entire edifice must be at least sketched out in advance in some detail, thus tending to direct funding to more clearly defined work and to methodologies where such an approach is possible. Applicants further may feel constrained to propose the focus of the research work to a topic they feel is more likely to be funded, and to using methodologies that they feel are more likely to lead to a successful outcome in the competitive application review process. Moving away from your mathematics example to health research, for instance, applicants may feel compelled to propose a randomized controlled trial to investigate an intervention, rather than using other approaches (e.g., a single case experimental design series) as randomized controlled trials are still perceived as the 'gold standard' even if other approaches may also have merit.

When funding such as the MacArthur Fellowship is awarded to a person, rather than a project, the person is free to pursue research in whatever way they deem best—rather than being constrained by having to pre-define their approach and filter this to maximise likelihood of being funded.