We know due to Doppler’s effect we see a red-shift in color of a star moving away from us.
Now consider a star that has only one planet revolving around it. As the planet revolves, it also moves the sun under influence of it gravity. Though very small effect, it does cause the star to wobble a bit in its position.
Modern day astronomy has become so accurate, that we can measure the minute difference in red-shift that can be caused by this wobbling. To the extent that we can measure the red-shift effect even in a walking star (walking speed is 4-5 km/hr).
And this is what astronomers are using to find stars that have planets.
Binary stars are a pair of stars revolving around each other. Now, a star that has a binary star will also wobble.
So, how do we know that a wobble is due to a plant and not a star? Simple, use method called trans. Watch the light carefully. If a binary star is so aligned that when revolving, it will pass between the star and us, the intensity of light of the star increases. But if a planet passes in front of it, the intensity drops. However small the drop is, the accurate observation detects that drop and identifies that as a planet.
We know that sun has solar flares. So might an others star? So, how do we know that the drop in intensity is not due to a flare at lower temperature (and hence less bright than the star surface). Simple, solar flares will not be as periodic as the planet’s revolution. Also, the “exact” intensity shift due to a flare is known.
In few years, Venus will pass between earth and sun and observational astronomers will use the observed drop in intensity to confirm the value the use in these experiments.
With this great tool in hand, they are now searching for earth like planet having a sun like stars. Once the find interesting planets, they will request time on Hubble to observe these planets.
Also check http://www.exoplanets.org/