Breaking boundaries in experimental physics
Witness the Big Bang
High energy experiments, made possible by particle accelerators, allow physicists to view phenomena that occur only in extreme conditions, like those at the beginning of the universe. These experiments can make particles that would be incredibly unlikely to appear on our planet today show up in experiment. With more energy in the beam, more exotic particles exist for longer, and may even exist long enough to be detected.
Achieving High Energies
Particle accelerators make these incredibly improbable events both likely and detectable, by sheer volume of interactions from the number of photons accelerated, and from the high energies that make unstable states more accessible. They allow physicists to confirm and learn about physics in conditions similar to the Big Bang, and confirm exotic predictions of particle physics.
A Powerful Microscope
Accelerated protons allow physicists to observe the particles that make up atoms at the very lowest levels. For something to be detectable in a microscope, it has to be larger than the wavelength of the photons hitting it. Wavelength is inversely proportional to momentum, so high-energy protons can see phenomena thousands of times smaller than any photon could detect.