Why are planetary systems so rare? According to this site there are 258 know planetary systems and 302 planets. Mostly each of the listed system has only 1 planet of Mercury's or Mars' size, while our system has up to 8 planets. From what I know our sun is not a "big" star, so theoretically other bigger stars probably threw out more planetary material, so they should have many more planets. 
Humans know about billions of stars and since planets are made from what remained from star's birth then why aren't there billions and billions of planets?
 A: Planets aren't very easy to spot - 258 known planets doesn't mean 258 planets out there.
There is a strong selection effect. Detecting planets based on the movement of a star could only detect very massive planets very close to the star which had a significant gravitational effect. Newer techniques based on a planet blocking some of the stars light are more sensitive to planets closer to the star - it would be very hard to spot pluto's effect at blocking the sun's ligth if we were a long way away
A: There may well be billions of planets in our Galaxy.
All stars in our Galaxy are in relative motion and occasionally a star passes very close to the line between observers on Earth and a background star. The light from the background star is deflected by the gravitational field in a phenomenon called gravitational microlensing. The gravity of a planet orbiting the host star can also deflect the light from the background star and thus the planet's presence known.
There has been a recent discovery by gravitational microlensing of a large population of "free-floating" jupiter-mass planets, or "nomads". These are planets which do not orbit a host star. The research suggests that there could be almost two jupiter-mass nomad planets for every star in the Galaxy. See http://arxiv.org/abs/1105.3544v1.   Other work suggests that there could be as many as 100,000 nomad planets (of masses ranging all the way down to almost Pluto's mass) per main sequence star, see http://adsabs.harvard.edu/abs/2012MNRAS.423.1856S.
A: That website hasn't been updated in over 2 years, and exoplanet discovery rates are always increasing. There are regularly updated repositories out there, for instance exoplanets.org, which is even used by professionals for getting quick access to whatever data has been published to date. (There are smart phone apps that draw on such sites, too, if you are interested). Note that there are at least 600 planets listed there.
In fact, there are likely to be many more planets in the data obtained by Kepler. Even though most of these have yet to be confirmed with followup observations, this article discusses how we do not expect many false positives.
And, as alluded to in @Martin's answer, there are biases in our methods of detection. The two most common methods today are transits and radial velocity, which are better at detecting close-in, large planets and close-in, massive planets, respectively. The various methods are summarized on Wikipedia. Exoplanet detection is still very much in the stage where the lack of a detection does not mean there is nothing there.
