It's not surprising that it took so long. The stars are so far away that even the light from our hundred nearest neighbors takes up to twenty years to get here. In addition, remember that we see Solar System planets by reflected sunlight. If we were 24 trillion miles away - the distance to the next star- even Jupiter would be submerged in the Sun's light.
This means astronomers don't find extrasolar planets simply by pointing telescopes at nearby stars. In fact, it was only in 2005 that we had the first image of an extrasolar planet. Out of nearly nine hundred identified extrasolar planets only a few dozen have been directly imaged.
So how do they find them if they can't see them? By detecting a planet's influence on its star.
The method which has enabled the discovery of the majority of extrasolar planets is Doppler spectroscopy, also called the radial velocity method or, popularly, the "wobble method."
Think how the note of an emergency vehicle's siren changes as it approaches and then passes you. When approaching, the vehicle overtakes the sound waves. This increases the frequency you hear, making the note higher. After it passes, the opposite occurs and the pitch drops. This is
The Doppler effect applies to light waves too. The spectrum of approaching objects is blue-shifted, i.e., it's detected at a higher frequency. Objects moving away are red-shifted.
Strictly speaking, planets don't orbit stars. There is a mutual gravitational attraction so that star and planet orbit their common center of gravity. This center is inside the star, so the interaction just makes the star wobble slightly.
If the orbiting planet causes the star to move alternately towards and away from us, a sensitive telescope may detect alternating blue and red shifts in the light spectrum. The frequency of the shifts shows the planet's orbital period and the size of the shifts tells us about the mass.
In 1995 Swiss astronomers Michel Mayor and Didier Queloz discovered the first extrasolar planet orbiting a sun-like star. It was quite a surprise, for it had at least half the mass of Jupiter, but was in an orbit closer than Mercury's is to the Sun. This was the first of the "hot Jupiters."
Hot Jupiters aren't the most common planets in the Galaxy, but they're the easiest to find. Massive and close to the star, their gravitational influence is maximized. And it doesn't take long for repeated observations to establish the orbital time. By contrast, Jupiter itself takes twelve years to orbit the Sun.
The second most common way of finding planets is the transit method. A transit happens when a planet passes across the disc of its star. This causes a tiny dip in the star's brightness.
The size of the dip provides evidence of the planet's diameter, and the orbital period is determined from the timing of the dips. Unfortunately, there are many causes for variation in starlight, so transits need to be confirmed by other methods.
NASA's Kepler mission, launched in March 2009, has been monitoring a small, populous star field. Not being subject to distortion by Earth's atmosphere, its sensitive photometer is able to detect the transits of smaller planets. One of the mission goals is to find earth-like planets. (Kepler's mission ended in 2013, but the data it collected is still being analyzed.)
There are images related to this article on my Pinterest board Extrasolar Planets.