Spectrometers generally produce a single spectrum from a light source.

A hyperspectral imager produces a spectrum for each pixel in the image. See https://en.wikipedia.org/wiki/Hyperspectral_imaging

Like with a grating I can see how you can obtain a spectral pattern for many wavelengths, but with a bayer array, surely you can only distinguish R, G, B.

Would I be right though in thinking, if there's no stray light and only the light from the phone screen shining on the apple, you can cycle through many wavelengths of light, enabling you to measure the reflected light to determine the spectra.

One thing I was wondering so pesticides can presumably also be detected using visible light spectroscopy then?

They inversed the principle: Rather than controlling the bandpass of the sensor (which a grating does passively, over the spatial domain) they are controlling the spectrum of the light source. You can produce a varying light spectrum by setting R/G/B on the display, and record the light reflected by the object with your R/G/B camera pixels, which also have a certain frequency response.

I have a feeling you can use this to recreate a coarse approximation of the full spectrum and then use it to infer some of the object parameters, like pesticides on fruit.