I have an answer to a slightly different question: If you're an academic, how can you impact the world? This is something I thought about a lot when I was an economics PhD student.

Here are some ideas (mainly geared towards economics and especially microeconomics):

• Work for an organization whose aims you support, like the IMF, World Bank, the Congressional Budget Office, or the Federal Reserve. One of my friends at the Fed says that this work may not be impactful on a day-to-day basis, but you can occasionally have an enormous impact -- for example, by helping the Fed quickly figure out how the pandemic should affect its actions.
• Basic research can change the minds of journalists and policymakers. Some of these people actively read newly released working papers, and some read economics coverage in the press. For an example closely related to my PhD dissertation: Chetty et. al.’s work on teacher value-added got a lot of attention in the popular press, as did similar work, and Chetty was called to testify in Vergara v. California, which (briefly) eliminated teacher tenure in California. Also, if you’ve been following discussion of economics in the popular press recently, you may have seen discussion of papers on the minimum wage and on immigration that are older now, but are still influencing how people think.
• Program evaluations may have an especially good chance of impact. For example, follow-ups on the Perry Preschool Project and Abecedarian Project have probably broadly increased support for early childhood education -- many people are vaguely aware that the cost/benefit analysis looks good. And without Miguel and Kremer’s work on deworming, deworming would probably be much less common today.
• You can do methodological work that will make other researchers more productive and accurate (or turn them away from researching questions that may be unanswerable). For example, you can develop more accurate formulas for standard errors, so people don’t overstate their confidence in results. Returning to the early childhood education example, more modern and careful analysis has made cost-benefit analyses look worse here.
• If you are already well-established as an expert, you can work in prestigious governmental positions like the Council of Economic Advisors and advise influential people. If you're curious about what it's like to work in such a position, I recommend Casey Mulligan's book.
• Your research may inspire you to found a charity like GiveDirectly, which was founded by economists.
• You can write for the popular press or blog to distribute ideas that are already well-known to academics.

If you're a grad student altruistically looking to make an impact, and you're not trying to get a top-notch academic job, it may be especially productive to focus on the sort of work that the job market does not reward, since those topics may be under-studied.

Edit to add a couple more thoughts: A lot of academic economists advise policymakers directly -- for example, Jonathan Gruber had a lot of influence on the Affordable Care Act, and a handful of my Harvard economics professors had met with presidents. Additionally, I have a sense that think tanks are pretty influential, but I don't know anyone who works for one or exactly how that works.

If you have not already done so, you may be interested in reading this report on Early Field Growth by Luke M. from the Open Philanthropy Project.

I know one of the examples I've heard of is neoliberalism and the Mont Pelerin society. You may be able to use that as a case study.

This may be less meta than you are hoping for, but may contain some useful advice/references: The dos and don’ts of influencing policy: a systematic review of advice to academics. Influencing policy is at least one way that academic ideas can travel to the wider world. 

I expect another is producing accessible content on the topic in question (e.g. writing popular blog posts, books, documentaries). It seems like these can sometimes be a catalyst for ideas becoming more widely known in the public. Examples of books that might have had or could have a broad impact are Animal liberation (Peter Singer), Silent Spring (Rachel Carson), Doing Good Better (Will Macaskill) or Human Compatible (Stuart Russel).