Most of the answers in this thread are at least partial nonsense.
For example, professionals do not typically open lenses up to f/1.2 to shoot headshots with full frame cameras, because depth of field at that aperture and range is so short that you'll have one eye in focus and nothing else! Trust me, I use these sort of lenses! Oh - and image quality is often relatively poor until you close down a couple of stops. And compromises to get large apertures mean that an expensive f/1.2 lens may not get as sharp as a lens with narrower maximum aperture, once closed down; possibly the sharpest standard lens there is the f/2.8 46 mm equivalent on Sigma's Foveon sensor compacts. That f/1.2 85 mm lens that someone above thinks make so very professional has rather poor edge quality until it closed down by 2 to 3 stops (for example, see photozone.de's review of the lens). Now, a real pro would know this — and then either buy the much cheaper and light f/1.8 version, or buy the f/1.2 if he needed the low light focusing boost.
On the other hand, the affect of going from f/1.8 to f/1.2 at headshot distances is provably insane — or at least something you'd very rarely want to do (DoF chart for 85 mm f/1.2). At 2.5 m, it's the difference between a 10 cm sharp zone and 8 cm one. Would any sane person really care about the 2 cm difference? You're more likely to want to narrow the aperture and increase DoF, rather than to double the cost for a lens that can reduce it by 2 cm!
The reason for huge lenses in YouTube videos is because they're often for Guys With Cameras — gullible middle aged men who want to become glamour photographers. The big zooms and huge DSLRs impress them because they're what they see paparazzi using on TV and because of the obvious reason. Real portrait photographers are likely to use something much more compact, like a Fuji X with a 56 mm lens, or a Sony A7 or Nikon whatever with a 50–85 mm.
Possibly the main use of VERY wide aperture lenses by professionals is not to shoot but to focus. Focus systems hate dim light, especially the phase detect systems on DSLRs. Amateurs, on the other hand, often spend hours taking pictures of a tree with just one leaf in focus thanks to that insanely sort depth of field — and if makes them happy, why not?
Another point to understand about lenses like the f/1.2 85 mm Canon is that their DoF is small at f/1.2 and headshot range that they often can't be focused accurately enough for a competent picture. This isn't a problem for mirrorless cameras, which focus much more accurately because they do so from the imaging sensor.
Most of all, the f/1.2 Canon 85 mm isn't sharper than the cheaper and lighter f/1.8, it's LESS sharp. This is almost always the case with very bright lenses on full frame - they're special purpose tools designed to sell either to serious shooters with very unusual needs (20% of the market?) and people who don't actually know what they're buying (80%?)
Again, this doesn't translate to other systems — f/1.2 is the equivalent of f/1.4 or f/1.8 for Fuji and Micro 4/3 systems. f/0.95 is the specialist/insane aperture for standard and short telephoto lenses.