Asteroids are tiny compared to their relatives, the larger terrestrial and gas giant planets. However, their importance to our solar syatem's evolution, their potential threat to Earth as impactors, and the fact that they are the keys to our next step in space exploration and development, render them perhaps the most important objects of study. There are three principle reasons that make asteroids essential objects of study:
Planetary Defense – Asteroids (and comets too) have been striking the Earth since the earliest history of the solar system, some 4.3 billion years ago. Although the frequency impacts from the largest and most dangerous asteroids has greatly diminished, the Earth is still struck by numerous small asteroids, and there remains a possibility that the planet could be impacted by large asteroids in the future (>500 meters diameter). One great incentive to survey and study asteroids is for humans to learn how to best deflect a potentially dangerous impactor.
Space Resources - Asteroids contain an abundance of minerals and elements required by humans to survive and evolve here on Earth, and to explore and develop space. Additionally, many asteroids are less energetically expensive and less costly (for spacecraft) to visit than even our moon. This is partly true because near-Earth asteroids, living up to their name, reside within orbits that periodically bring them to us, making them attractive targets for mineral exploration and extraction. There are already well-funded private organizations formed that have begun the technical steps to explore and ultimately extract resources from near-Earth asteroids.
Science – Asteroids, despite being smaller in size than their relatives the larger planets, comprise a significant structural component of our solar system. Their size, shape, composition and orbital families contain information about the past structure of the solar system, how it evolved through time to appear as it does today, and how it is further evolving to affect other solar system bodies. Understanding the principle of solar system formation and planetary evolution contributes towards our understanding of how life evolved and how it's been affected by impact processes.