Exoskeletons are formed by layers of chitin formed from the epidermis. While most invertebrates get their colouration from pigments and reflect light like so: 

Inverts that have shiny exoskeletons get their colour and shine from the way the light interacts with the microscopic structures of their exoskeletons. The light that hits these structures is specular reflection, while light hitting normally pigmented exoskeletons is diffuse reflection (like above with the red lady beetle).

For example: 

Chrysina gloriosa

These beetles get their shine a liquid crystalline material produced on the external surface that assembles into three complex polygonal shaped structures: hexagons, pentagons and even heptagons. They selectively reflect left circularly polarized light on the beetles excoskeletons that gives that shiny and brilliant green appearance!

• Here’s the research on this beetle and it’s shiny extoskeleton
  

Another example (another couple of coleoptera) Chrysina aurigans (gold) and Chrysina limbata (silver).

These beetles get their shiny excoskeletons from multilayer nanostructures allowing for that metallic shine appearance. Also to note these beetles are not  iridescent! (this means they appear gold or silver from every angle). 

These beetles exoskeleton’s are made up of over 70 layers of chitin and each have difference refractive indices that allows light waves to be reflected at each interface. 

• Read the full research paper here
  

Another example is the use of  iridescence that can create that shine in excoskeletons (more beetles I know. But beetles are the largest order and have lots of good examples). 

(a) Loxandrus rectus, (b) Phalacridae gen. sp., © Cicindela scutellaris scutellaris, (d) Amarygminae gen. sp., (e) Phanaeus vindex, (f) Eupholus sp.

There are a number of structures and mechanisms that these beetles use to get their shiny exoskeletons.

• Multilayer reflectors - that are thin layers of chitin that have difference refractive indexes and are spaced to one-quarter of the visible light wavelength. 
• Additive or ‘pointillistic’ colour mixing - where the external surface of the beetle is covered with an array of closely packed, minute punctae. That each individually reflect narrow bands of wavelengths 
• Circularly polarizing reflectors  - Like explained already above in Chrysina gloriosa
  
• Broadband reflectors - these are multilayer structures that reflects most or all wavelengths of light simultaneously
  
• Three-dimensional photonic crystals -  these crystal structures (found in scales on the exoskeleton) have closely-packed shapes; hexagonal, diamond lattice, pentagonal, ect, that reflect saturated colours and reduce the angle dependency. Colour is produced by the high ordered lattice in the scales. here’s what they look like (as shown in Pachyrrhynchus congestus pavonius).

• Read more about that research on iridescence mechanisms here