Do plants need sleep?
The quote as phrased is a bit non-sensical really because it mostly hinges on what you define as 'sleep' or 'rest'. Plants, like most living things, including us, are far from dormant at night.

It's true that diurnal patterns govern the behaviour of most, if not all, living things that are exposed to them (1). Plants are no exception to this.

For example, photosynthesis is a 2-component process. With the so called 'dark' and 'light' reactions of photosynthesis (more correctly "light-dependent" and "light-independent"). Though both happen during the day and night, it is at it's peak during peak sunlight.

In CAM plants, to avoid dessication, there is differential carbon fixation because the stomata are kept shut during the hot daylight hours.

Respiration however, where the produced food is converted to energy, primarily happens at night.

The Khan Academy has a great 'walkthrough' of the steps of photosynthesis:
https://www.khanacademy.org/science/biology/photosynthesis-in-plants/the-light-dependent-reactions-of-photosynthesis/a/light-dependent-reactions

(1). The diurnal course of plant water potential, stomatal conductance and transpiration in a papyrus (Cyperus papyrus L.) canopy
https://link.springer.com/article/10.1007/BF00379885