It is important not to think just in terms of "the plants", as in organic gardening more of what we do revolves around the soil and the microlife that should be thriving within it. That said, like other living organisms plants need B vitamins for cellular metabolism- turning carbohydrates (a result of photosynthesis, in the case of plants) into energy used to build proteins and cell structures. Plant cells require other vitamins as well, like humans do. However, unlike humans plants can actually synthesize all of the vitamins they need. They build vitamins along with most of the other things they require from the basic elements (carbon, nitrogen, phosphorous, potassium, calcium, magnesium, sulfur and trace elements) obtained from the soil and atmosphere.
So while plant cells really do require vitamins such as the B vitamins, they don't need us to provide them under typical growth conditions. B vitamins at low doses may hasten seed germination and early development, and they also may help during transplants to prevent shock. Growth hormones, particularly auxins are good for these purposes also, and may be in products combined with one or more of the B vitamins.
Beyond the plants; those micro-organisms living in the soil also require B vitamins for cellular metabolism and an abundance of B vitamins in the soil (especially B12) may lead to a boost in productivity and growth. Anything that benefits the microbes should result in happier plants. I wouldn't go out of my way to provide vitamins, though. Human B-complex tablets are probably unsuitable for horticultural applications, just because the dosages they contain may be toxic to microbes or plants and there are other ingredients within tablets.