(The Telegraph misrepresents/misunderstands Rees positions - he is a good astrophysicist, and merely discusses the past debate in his book. But what else do you expect from a tabloid?)
The concern about microscopic black holes started a few years ago ($\sim $2002) because theories with large extra dimensions suggested that gravity could have different strength over short distances, and this might allow black holes to form with far less energy than if you had to get it inside the Schwartzschild radius with the normal 3D gravity. That would be yummy for particle physicists, who would get interesting jet patters in the LHC if this were true. This was serious theoretical physics, although by now the evidence has turned against it.
Of course some people started to worry that this might harm the Earth (together with other exotic risks such as vacuum decay or stable strangelets). Eventually, in the face of lawsuits the physicists actually did some proper risk assessments and came up with some good arguents for why things are safe. Each of these arguments are somewhat weak, but together they bound the risk rather well.
The most popular argument is the "cosmic ray argument": Earth has been hit with intense cosmic rays for eons, and this has not produced any black hole that eats the planet. Hence the LHC is safe.
This is a nice argument but has a few flaws. One is observer selection effects: if Earth had been imploded, we would not be around. We can only evolve on non-imploded planets. However, the argument is easily saved by looking at the moon or more remote planetary bodies: the lack of imploded planets in the solar system (or the current low supernova rate) still gives evidence. One can also use the age of Earth as a bound on the risk per year to $<10^{-9}$, which works against a large category of risks, including cosmic ray black holes.
Another problem is that maybe ray colisions produce remnants with significant momentum that escape into space, while head on accelerator colisions have near zero momentum and fall down. Dar et al. analysed how at least strangelets are slowed down among cosmic rays and then their risk can be bounded by observed survival of neutron stars and white dwarfs, plus the supernova rate.
The big analysis of black hole generation and safety is due to Giddings and Mangano. They also point out that dense stellar remnants would tend to gather black holes from cosmic ray hits and would implode at a very noticable rate if the risk to Earth was even tiny. They estimate the accretion ability is too low to matter over billion year timescales. Plus, of course that Hawking radiation will get them.
In summary, the risk looks like it is acceptable (indeed, the lack of black hole production so far makes it look like it is actually zero). But the concern was based on real theories in physics, physicists did not initially take public concerns seriously enough, and when they did risk assessments they had to do far more work than they expected to bound the risk. Given the importance of avoiding low-probability disasters with huge impact (not to mention existential risk) this ought to be a lesson for future projects to do the risk assessment well in advance.
[ A simple classical calculation ignoring Eddington luminosity: if we have a black hole of mass $m(t)$ it will have cross-section $27\pi R_S^2/4= (27\pi G^2/c^4)m(t)^2$. If it just sits in an environment of density $\rho$ it will grow as $$m'(t) = (27\pi G^2/c^4)\rho v m(t)^2 = A m(t)^2$$ where $v$ would either be its velocity or the speed of sound in the surroundings. If we set $\rho=$13,000 kg/m$^3$ and $v=$11,000 m/s $A=6.6888\cdot 10^{-45}$. The solution to the equation is $$m(t) = \frac{1}{C-At},$$ where $C$ is an integration constant we set to $1/m(0)$. This will diverge as $t=1/m(0)A$, roughly corresponding to the hole gobbling up the planet. So we find that a one kilogram black hole takes $10^{36}$ years to matter. You need to start with nearly an Earth-sized black hole to have near-future disasters. The case for multidimensional theories is done in Giddings and Mangano. ]
