First of all, just to be clear (some would say “needlessly pedantic”) time doesn’t slow down for Observer A close to c; he (or she) will experience time passing at the rate within his own reference frame regardless of speed or acceleration. Observer B in an inertial reference frame outside will perceive time to be moving slower in the accelerated frame than in his own surroundings, and Observer A inside the accelerated frame will perceive things outside to be going really fast, per the Lorentz transformations.
I have to admit I don’t know what the presenter means by “the singularity would cease to exist,” however, once Observer A has passed the event horizon–the point at which the intensity of the gravitational field is so strong that no possible trajectory can escape–we can’t really say anything useful about what happens to him. Hypothetically, Observer A will perceive things outside of his frame to happen faster and faster while in fact being subject to relativistic transformation will be moving slower and slower, incrementing toward complete motionless (inside the frame) as he approaches c; indeed, aside from the fact that all incoming light is going to be compressed and blueshifted into X-ray or gamma ray range, frying him to a crisp, he’ll be able to see the end of the universe in the blink of an eye. And if we take a rotating black hole (which most if not all most likely are) you have an additional zone called the ergosphere, in which relativistic frame dragging–warping of space–occurs. This can result in some really funny results in which, from the perception of external Observer B, Observer A goes faster than light. He’s not, really; it’s just that the space through which he travels is being moved (which can occur at any speed), and when you add all of that together you can get velocities that are greater than c in flat space, which the area around a large massive object (especially a rotating one) is not. However, the reality is that with any singularity smaller than a galactic mass the tidal forces from extreme gravity gradients would shear Observer A into component atoms long before he got around to observing anything cool.
This segues into a direct answer to your question; light is massless, and is therefore permitted to move exactly and only at c per the tenets of special relativity. It also travels exactly in straight lines until it hits something. (A quantum field theorist would qualify this by saying that light takes all paths but it all averages out to follow a straight line, but they’re kind of an indeterminate bunch anyway.) However, these “straight lines” (physicists call them geodesics) lie upon the plenum of spacetime which is distorted by the presence of mass (which is just energy bound up in a particular form) and some other, more exotic forms of energy which may or may not actually exist. The conservative energy field created by the presence of mass is, of course, gravity, and so a concentration of gravitational energy will cause space to be distorted and thus light to bend around it, just as a marble will roll toward a bowling ball dropped in the middle of a mattress. (For those playing along at home, it is inadvisable to store your bowling ball in the middle of your bed mattress; this is only done by professionals for the purpose of illustration and should not be repeated without adult supervision.)
The set of equations that relate the presence of energy to the distortion of spacetime, and thus would describe the geodesic lines that light would follow, are called the Einstein Field equations, named after the eponymous scientist despite being independently developed by mathematician David Hilbert and his students. (To be clear, Einstein developed the fundamental concepts for GR, but Hilbert et al developed a great deal of the underlying math. Hilbert himself credited Einstein with the essential discoveries, while according himself only nominal credit for his work.) The EFEs are a descriptio of relations between a set of four symmetric tensors (stress-energy, Ricci, curvature, and the metric tensor); this is obviously somewhat beyond high school mathematics, and I’ll defer to Chronos or another professional physicist working with relativity theory to explain the relations in more detail, partially because I don’t want to trod on anyone’s prerogatives but mostly because I don’t have a modern physics text here at work and don’t want to futz up details in an area I haven’t studied in over a decade.
In reference to your final question regarding light emitted from an object moving at c, all light moves at c (though its local frame) regardless of the speed of the frame it was emitted from. This seems counterintuitive from normal experience, because when we through a baseball from a moving train we are told to add the velocity of the train to the velocity of the ball so that we can figure out just how fast it is going when it smacks some poor bystander in the head. (I sincerely hope that these types of problems are only performed in classroom problem sets.) Since this doesn’t happen with light, but we’re still stuck with having to deal with conservation of momentum and conservation of energy, we have to resolve this by changing the frequency of light (which is directly related to the energy of an individual photon). So your photons emitted by an object moving at close to c would be redshifted into oblivion, and photons emitted by an object moving at c would be entirely hypothetical, as they’d have a frequency of zero. This also ofters the conceptually problematic issue that if you are moving faster than c, the way to slow down is to shoot photons out the back, which is kind of like hitting the accelerator petal to slow down, something that Audi 5000 owners know to be distinctly ill-advised.
Here is a pretty clear nontechnical reference from our friends at Caltech on the fundamentals of general relativity and gravity, and here John Baez gives a layman’s overview of the Einstein Field equations. Enjoy.
Stranger