// started 10/04/2014, added remark 17/4/14
The paradox, as put forth by the physicists Enrico Fermi and Michael H. Hart, is the contradiction:
The apparent size and age of the universe suggest that many technologically advanced extraterrestrial civilizations ought to exist.
However, this hypothesis seems inconsistent with the lack of observational evidence to support it.
Wikipedia has a very good article  and mentions many possible solutions to the paradox, to which I would like to add my own:
I have the feeling that the idea of "space exploration" - meaning the discovery of our universe - via physical presence (either via probes or even humans) might be typical for 20th century thinking. I believe this idea of exploration might not be applicable for the future and instead we will rely on discovery through simulation.
By that I mean; given enough computing power on earth, and a deeper understanding of our reality (eg. physics) it will be possible to compute any future (and past) state of the universe, given enough computation time. Hence I would argue that highly advanced civilizations will not be required to travel anywhere outside their home world (given that no interstellar accident, eg. supernova, occurs), because any corner of the universe can be simply computed. Obviously it might be unlikely to simulate every location at every time in the universe, but still, enough computing power might be available to discover remote species. Occasionally a few probes might be sent out to actually confirm the simulations, but those would be a rare occurrence. furthermore, such an advanced species might go silent on purpose and instead only rely on listening for their simulations, rather than sending out any traceable signals of their own.
hence any civilization would go through a phase, like the one we are in right now, where an abundance of radio signals is transmitted. however this phase might be very brief and it would be a very rare occurrence for two civilizations to exist within hearing distance and at about the same stage of development.
furthermore, in such a setting, any conflict between advanced species might be futile, because everyone is always aware of any action of any another species, ie. they are all in a state where the universe holds no more secrets. Of course this implies that the universe is somehow deterministic in nature or at the very least strongly convergent in the probabilities for many events.
Seth Loyd (professor for mechanical engineering at MIT) first propagated the idea (30 years ago, according to ) that quantum entanglement would equilibrate particles over time. While his work is still outside the realm of mainstream physics theory, it seems to me as a possible corner stone for the universal simulation outlined above. In his theory (from what I understood as a layman) he describes that the quantum state of particles (ie. their spin) becomes increasingly correlated over time. Whether this entanglement is somehow dependent on location is not clear to me at this point (as well as lots of other points). Still it hints towards a strong convergence in probabilities of the universe. As mentioned in , what we perceive as reality might just be the most probable state particles tend to take, given their current state of entanglement with each other.
For the interested reader, further research is available under  and . Added 10/7/2014: a very humble start from 1984: http://www.newyorker.com/online/blogs/elements/2014/07/the-video-game-that-maps-the-galaxy.html