An insectivore is a type of carnivore with a diet that consists chiefly of insect

Insect

Insects are a class of living creatures within the arthropods that have a chitinous exoskeleton, a three-part body , three pairs of jointed legs, compound eyes, and two antennae...

s and similar small creatures. An alternate term is entomophage, which also refers to the human practice of eating insects

Entomophagy

Entomophagy is the consumption of insects as food. Insects are eaten by many animals, but the term is generally used to refer to human consumption of insects; animals that eat insects are known as insectivores...

.

The first vertebrate insectivores were Amphibian

Amphibian

Amphibians , are a class of vertebrate animals including animals such as toads, frogs, caecilians, and salamanders. They are characterized as non-amniote ectothermic tetrapods...

s. When they evolved, 400 million years ago, the first amphibians were piscivores, with numerous sharp conical teeth, much like a modern crocodile. The same tooth arrangement is however also suited for eating animals with exoskeleton

Exoskeleton

An exoskeleton is the external skeleton that supports and protects an animal's body, in contrast to the internal skeleton of, for example, a human. In popular usage, some of the larger kinds of exoskeletons are known as "shells". Examples of exoskeleton animals include insects such as grasshoppers...

s, thus the ability to eat insects is an extension of piscivory.
Although individually small, insects exist in enormous numbers - they number over a million described species and some of those species occur in enormous numbers.