The most common interpretation is that these theorems prove that at the Big Bang and in the astrophysical collapse of a black hole, we really do get densities as extreme as the Planck density.
I don't think any professional physicists believe that the process of forming a singularity proceeds as described in the singularity theorems to densities beyond the Planck density. It's widely accepted that under these conditions, quantum effects are strong and we can't trust classical GR.
There is a little more wiggle room for the idea that the density might stop short of the Planck density. There is a field of semiclassical gravity (e.g., Visser 2009), and if you take their techniques seriously, it's possible that in the astrophysical collapse of a black hole, classical GR fails far below the Planck scale. I think very few relativists or people working on quantum gravity would be willing to bet a six-pack that this is right, because there are serious questions as to the validity of the techniques used in semiclassical gravity. (They have to do some pretty funky renormalizations, and there is no contact with experiment.)
The short answer is that the "singularity" of which you speak is a mathematical artifact that only arises under certain conditions, and cannot meaningfully be said to exist. A black hole consists of an event horizon and nothing else.
The first sentence is reasonably accurate, if you keep in mind that there is indeed something very, very dense inside a black hole, and the early universe was indeed very, very dense. The second sentence sounds silly, although maybe it would be less absurd if put in context. Mass-energy is locally conserved in GR, so the mass of the collapsing body has to be in there somewhere.
Visser, "Small, dark, and heavy: But is it a black hole?," http://arxiv.org/abs/0902.0346