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Freshly moulted katydid eating its exoskeleton.

Quote from en.wikipedia.org/wiki/Exoskele…
Exoskeletons contain rigid and resistant components that fulfil a set of functional roles including protection, excretion, sensing, support, feeding and acting as a barrier against desiccation in terrestrial organisms. Exoskeletons have a role in defense from pests and predators, support, and in providing an attachment framework for musculature.
Exoskeletons contain chitin; the addition of calcium carbonate makes them harder and stronger.

Ingrowths of the arthropod exoskeleton known as apodemes serve as attachment sites for muscles. These structures are composed of chitin, and are approximately 6 times as strong and twice as stiff as vertebrate tendons. Similar to tendons, apodemes can stretch to store elastic energy for jumping, notably in locusts.

Since exoskeletons are rigid, they present some limits to growth. Organisms with open shells can grow by adding new material to the aperture of their shell, as is the case in snails, bivalves and other molluscans. A true exoskeleton, like that found in arthropods must be shed (moulted) when it is outgrown.[5] A new exoskeleton is produced beneath the old one. As the old one is shed, the new skeleton is soft and pliable. The animal will pump itself up to expand the new shell to maximal size, then let it harden. When the shell has set, the empty space inside the new skeleton can be filled up as the animal eats.[5] Failure to shed the exoskeleton once outgrown can result in the animal being suffocated within its own shell, and will stop subadults from reaching maturity, thus preventing them from reproducing. This is the mechanism behind some insect pesticides, like Azadirachtin.