In my opinion, if something's worth understanding, then it should be understandable by someone with no formal understanding of economics.
The area of economics that studies how to design platforms is called market design. Al Roth and Lloyd Shapley recently won the Nobel prize for their work in this area. Here are some real world examples of market design in practice:
The National Resident Matching Program uses the deferred acceptance algorithm, developed jointly by Gale and Shapley, to match medical students to residencies.
Buying and selling kidneys is illegal, but economists have been instrumental in developing algorithms tools to match recipients and donors. See e.g. here.
Google decides which ads to show using techniques developed by economists, here is Google's chief economist on the topic.
The US and other developed countries used to allocate wireless spectrum to telecommunication firms through committees or lotteries. The system was widely perceived to be unfair and manipulable by powerful private firms. Economists have been instrumental in developing spectrum auctions which allocate spectrum efficiently, as well as raise revenue for governments.
Economists design prediction markets to try to provide forecasts of future events. Empirical evidence suggests that such markets regularly outperform expert forecasts by harnessing the 'wisdom of the crowds'.
These are just a few examples that come to mind. Economists help design platforms for school choice, dating services, and major technology companies like amazon or google. Al Roth wrote a pop economics book on the topic of matching markets called Who Gets What, And Why?.