Metals are shiny because of the light re-emitted (or reflected) by the vibrations of free electrons present on the surface of the metals.
When light falls on the metal surface, the photons of light are absorbed by the free electrons and these electrons move from one energy level to higher energy level. As the electron again drops back to lower energy level, it emits energy in the form of photons. This movement of electrons from lower to higher energy levels and vice versa causes electrons to vibrate at the atomic level. The wavelength of energy emitted by the electrons makes the metal shiny. Different metals emit different types of wavelengths. For example, copper and gold emit light in the wavelength range of 400-700 nm which gives a yellow shiny appearance to the metal. This shiny appearance of the metal surface is known as luster.
Note that metals can reflect only visible and infrared light, not UV light. This is because the light with a specific frequency falling on the metal surface causes the electrons to vibrate with the same frequency. The electrons can match the low frequency of visible and infra light and thus reflects the light. But in the case of UV light which has high frequency, electrons cannot match the frequency, and thus, metals become transparent to UV rays. For example, aluminum and silver become transparent to light having wavelength below 90 nm and 140 nm respectively.
This is the reason why we get sunburn even if we take shelter under the shadow of the metal roof made up of aluminum. The metal roof reflects the visible light but becomes transparent to UV light.
In simple terms, the light reflected by the electron cloud present at the surface of the metal makes the metal shiny. Due to this shiny property, metals such as silver and gold are widely used for making jewelry. Mirrors are also made by spraying a thin layer of silver or aluminum on the back of the glass sheet due to their high reflective property.
You might have noticed that usually, the metals with smooth surface appear shiny because a rough surface causes light to reflect diffusely. Therefore, for metal to look shiny, it should be polished regularly to remove the layer formed over the surface due to reaction with the environment. This is why the metals appear dull after some time and lose their luster. Polishing the surface can help metal to regain its luster properties. This dull layer on the surface of the metal can also be removed by chemical treatment. For example, the Gold and Silver jewelry with complex designs cannot be polished by ribbing its surface. To make these design jewelry shiny again, chemical treatments such as jewelry cleaner solutions can be used.