Why Detect Them?
The gravitational waves that LIGO can detect are caused by some of the most energetic events in the Universe—colliding black holes, merging neutron stars, exploding stars, and possibly even the birth of the Universe itself. Detecting and analyzing the information carried by gravitational waves is allowing us to observe the Universe in a way never before possible. It has opened up a new window of study and has already given us a deeper understanding of cataclysmic events, and ushered in cutting-edge research in physics, astronomy, and astrophysics.
Historically, scientists have relied almost exclusively on electromagnetic radiation (visible light, X-rays, radio waves, microwaves, etc.) to study objects and phenomena in the Universe (some are trying to use subatomic particles, called neutrinos, as well). Each of these sources of information provides scientists with a different but complementary view of the Universe.
Gravitational waves, however, are completely unrelated to electromagnetic (EM) radiation; They are as different to EM radiation as hearing is to vision. Thus, they are unique messengers of information about cosmic events. Having this new 'sense' with which to observe the Universe is important because things like colliding black holes are utterly invisible to EM astronomers. To LIGO, however, such events are beacons in the vast cosmic sea.
More importantly, since gravitational waves interact very weakly with matter (unlike EM radiation), they travel through the Universe virtually unimpeded, giving us a clear view of the gravitational-wave Universe (instead of X-ray vision, it's gravitational-wave vision!) The waves carry information about their origins that is free of the distortions or alterations suffered by EM radiation as it traverses intergalactic space.
With this completely new way of examining astrophysical objects and phenomena, LIGO has opened a new window on the Universe, providing astronomers and other scientists with their first glimpses of previously unseen or unseeable wonders.