I read "Plants and the human Brain" by David O. Kennedy recently, it explains how plants release a cocktail of volatile signaling compounds, kind of like plant "pheromones", whenever leaves, stem or roots get cuts or bruises. This start a cascade signaling between the plants own leaves as well other plants nearby, neighbors that often doesn't even need to be related species. Plants got ways of "learning", they got cell receptors and they got complex hormonal type of communication but, lets be careful about calling it pain. its more like discomfort. When plants are fed on, which they are very used too, they respond by balancing replacement of lost tissue and producing feed detergents against the attacking invertebrates. Plants remember what works against specific attacking pests, but keep in mind that pain is very expensive physiologically, especially maintenance. If we look at insects they keep their neural system to minimum because it drain staggering amounts of energy, they also don't have pain since it doesn't improve survival any more than the same type of epigenetic memory they share both plants and us. Plants spend a lot of energy on producing chemicals, especially for defense against UV and insects, analyzing plant tissue for chemicals takes immense amounts of work because there is no end to the amount of different chemicals each of them produce.
That video also forgot to mention that all life is electric and that receptors are all build on the same framework, the g-protein coupled receptor, when signals are relayed from one cell to another there often a small electric pulse involved if the signal need to travel fast. Photosynthesis is also "electric". Thats why ether and chloroform will affect a lot of organisms and why trace amounts of hormones in human urine can boost root growth at the plants we piss on.
Sometimes I find it annoying when popular science has to oversimplify things to the point they are skipping a lot of the interesting details, just to write something sensational.