Our cast of characters: Service A, Person 1, and Person 2.
Person 1 makes an API request to Service A. It's done over HTTPS, and Person 1 trusts that the response received is genuine, because it came over HTTPS, and certs were verified and so forth.
Person 1 sends the text of the API response to Person 2. Let's say Person 2 is a journalist, and Person 1 is a potential source. Person 2 doesn't know Person 1, and so logically doesn't blindly trust them. But Person 2 wants to report on some factual thing about the API response. How can Person 2 be sure that Person 1 hasn't modified the API response before Person 2 receives it?
If the API response is signed somehow, Person 2 could go retrieve Service A's public key (directly from Service A) and independently validate that the API response was actually received from Service A, and that it hasn't been tampered with.
I'm not sure I agree with the original author that this is a critical problem that needs to be solved, but that's the gist of it.