This is known as the instrumentalist approach to quantum mechanics, where we deliberately avoid thinking about the philosophical interpretations of things like the wavefunction, and only focus on the results of experiments. The thinking being, the observables don't change regardless of whether you pick MWI or pilot wave or the Copenhagen interpretation, so what does it matter? (The actual phrase "shut up and calculate" is older than this paper; I think David Mermin coined it.)

This approach rubs some people the wrong way, either because it's seen as a failure of imagination on the part of physicists to look at the bigger questions, or because it drifts close to the discredited logical positivism of the mid-20th century. Those are valid complaints, but I think it can also be seen as humility on the part of physicists, and an understanding that this is outside their field of expertise. "Science" is a fundamentally empirical exercise; if you're doing thinking without empiricism, it might still be valuable thinking, but it isn't science. "Shut up and calculate" is shorthand for, "Because these questions are not empirical, our tools and methods we have as scientists are not capable of addressing them." It's acknowledging that their powers are limited.

(I do think it would be nice if more physicists thought about it anyway, because in their abscence the void has been filled with woo by a bunch of New Age cranks, but that's not a moral obligation on their part)