To be clear, they don't sense "pain". Virtually every organism has receptor proteins that mediate molecular responses to stimuli. In plants, this might be a molecule from a pathogen, a mechanical thing (like an insect biting a leaf), or a molecule associated with damage (cell wall components or energy molecules being found outside where they're meant to be). Bacteria might excite their flagella to move away from danger, plants might isolate their cytoplasm to restrict pathogens, but these are well-documented physiological pathways only. Nobody is arguing that bacteria are capable of pain.

Only animals with a nervous system have a sense of "suffering" or "pain", and sure enough they behave a lot like humans in that respect too (by reacting with a response such as fleeing, fighting, screaming, etc). We have physiological responses too, but plants are a lot more like bacteria than animals in that sense. Having receptors and hormones is quite different to having a complex nervous system.