They do cite quite a few articles that show negative effects of specific vitamins.
The BBC article is merely clickbait.. People love to jump on the 'vitamins are actually dangerous' bandwagon. if you do a bit of digging into the studies they're quoting you'll find that their claims are unjustified.
The problem is that a lot of multivitamins (and supplements in general) have doses which are all over the place, and megadoses are quite common. You'll also find many reviews and testimonials that swear to the efficacy of any given dose of these supplements. On top of that, many doctors recommend that people take "a multivitamin", without any further guidance as to the ingredients or their dosage. Finally, a lot of people don't trust doctors and believe in alternative medicine, naturopathy, or what have you, from which they'll often hear recommendations to megadose because allegedly, the RDA's are set way too low (for a variety of reasons, depending on the supplement).
There's a lot of conflicting medical advice out there, and for the average consumer (who may have a host of medical issues, or suspected issues) it's not always as clear as "just take the RDA".
It's not a stretch that there are other correlations at play here. People who feel sick or lethargic try vitamins.