A company's data is like a raw set of food ingredients - they harvest that data, and collect it in one or more places.
An API is like the restaurant menu - it presents you with the ways that the data can be prepared for you, based on the skills, experience and methods that the company has available.
For some of the menu items, you may be able to make specific requests (no dairy products, for example) - whereas other menu items may be more static.
If you had the raw data yourself, and seasoned chefs, there's a chance you could prepare the meals that you really want, regardless of whether they're available in the API offered to you.
This analogy isn't perfect and isn't worth trying to apply in-detail, but for a high-level non-technical audience it could work.
(an example flaw: paging through search results is a common API operation that wouldn't really have a sensible restaurant analogy. at best, ordering profiteroles from the kitchen one at a time, each time requesting the least recently prepared one, or something like that)
