Can anyone explain the reason why CO$_2$ increases global temperatures (not the simplistic greenhouse analogy provided for public consumption)?
Borrowing an explanation from one of my other answers, the basic mechanism of the greenhouse effect is roughly as follows (note this is also a simplified model)


  The Earth is in (to all intents and purposes) a vacuum, so it can only
  gain or lose heat via radiation. The sun emits most of its radiation
  at visible and UV wavelengths. The Earth's atmosphere is fairly
  transparent at these wavelengths and so the Sun's radiation mostly
  passes through it and hits the surface. Some of this radiation
  (determined by the Earth's albedo) is reflected from the surface back
  out into space, but the rest is absorbed by the surface, which causes
  the surface to be warm. The surface loses heat by radiating in
  infra-red wavelengths. Greenhouse gases absorb some of the IR
  radiation, which causes the atmosphere to warm up (the GHG molecules
  transfer some of this heat to non-greenhouse gasses by collisions, but
  heat is also transferred upwards by convection). The warm atmosphere
  re-radiates some of this energy both upwards into space and downwards
  back to the surface. The part that is radiated downwards is also known
  as "back-radiation" (and is directly observable). Now the important
  factor is not the amount of outbound IR radiated from the surface that
  is absorbed, but the altitude at which there are not enough greenhouse
  gases above to absorb the IR radiated upwards from that layer, so that
  it can escape out into space. The lapse rate means that the
  temperature of the atmosphere decreases with increasing height. This
  means that the more CO2 we put into the atmosphere, the higher this
  emitting layer becomes, and the colder it is. As the amount of IR
  radiated depends on the temperature of this layer, if this height
  increases then the amount of IR radiated from the planet falls,
  leading to an energy imbalance, with the planet absorbing more of the
  sun's radiation than it emits as IR, and so the planet warms up. This
  continues until the radiating layer warms up enough for the outbound
  IR to be in balance with the incoming radiation from the sun. So the
  more CO2, the warmer the mean surface temperature, all things being
  otherwise equal.


Thus it is irrelevant that most IR emitted by the surface is absorbed by the atmosphere, it is the height from which IR is not absorbed that matters.  If it is colder than the surface, it will emit less IR than the surface is emitting and hence less energy is radiated into space.

For a more detailed explanation of why absorption at the surface is of little relevance, see this RealClimate article by Spencer Weart.

Just to add a historical note, this explanation goes back at least as far as Ekholm's paper of 1901 "On The Variations Of The Climate Of The Geological And Historical Past And Their Causes":


  The atmosphere plays a very important part of a double character as to the
  temperature at the earth’s surface, of which the one was
  first pointed out by Fourier, the other by Tyndall. Firstly, the
  atmosphere may act like the glass of a green-house, letting through
  the light rays of the sun relatively easily, and absorbing a great
  part of the dark rays emitted from the ground, and it thereby may
  raise the mean temperature of the earth’s surface. Secondly, the
  atmosphere acts as a heat store placed between the relatively warm
  ground and the cold space, and thereby lessens in a high degree the
  annual, diurnal, and local variations of the temperature.
  
  There are two qualities of the atmosphere that produce these effects. The 
  one is that the temperature of the atmosphere generally
  decreases with the height above the ground or the sea-level, owing
  partly to the dynamical heating of descending air currents and the
  dynamical cooling of ascending ones, as is explained in the mechanical
  theory of heat. The other is that the atmosphere, absorbing but little
  of the insolation and the most of the radiation from the ground,
  receives a considerable part of its heat store from the ground by
  means of radiation, contact, convection, and conduction, whereas the
  earth’s surface is heated principally by direct radiation from the sun
  through the transparent air.
  
  It follows from this that the radiation from the earth into space does 
  not go on directly from the ground, but on the average from a
  layer of the atmosphere having a considerable height above sea-level.
  The height of that layer depends on the thermal quality of the
  atmosphere, and will vary with that quality. The greater is the
  absorbing power of the air for heat rays emitted from the ground, the
  higher will that layer be, But the higher the layer, the lower is its
  temperature relatively to that of the ground ; and as the radiation
  from the layer into space is the less the lower its temperature is, it
  follows that the ground will be hotter the higher the radiating layer
  is.”


 [Ekholm, 1901, p19-20]

(h/t Steve Easterbrook's blog article)