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You can think of the phenomenon described by the formula a little like the sonic boom of an airplane moving faster than the speed of sound. If a charged particle hits a medium where it is traveling faster than the local speed of light, radiation with the properties described by the Frank-Tamm formula is produced. It is created by atoms that are polarized by the charged particle whizzing by. If it travels slowly enough, these light waves interfere and cancel each other out. With faster-than-light particles, however, this no longer works – and Cherenkov radiation is produced.
Luminous nuclear power plant
You can also observe them, for example as a blue glow in the cooling ponds of nuclear power plants. The radioactive fuel elements release fast electrons that can travel faster than light in the cooling water.
Cherenkov radiation is also generated naturally. Cosmic radiation from space triggers nuclear reactions in the Earth’s atmosphere, creating showers of particles that move extremely fast, sometimes faster than the speed of light. They produce “Cherenkov flashes”, which cannot be seen with the naked eye, but can certainly be detected with a suitable telescope.