Why are insects classified as animals?

All eukaryotic lifeform (i.e. not bacteria or archaea) are grouped into four kingdoms: Plants, Animals, Fungi and Protists.

A plant is a truly multicellular organism with cell walls composed of cellulose. They are typically autotrophic, meaning that they produce their own food, in this case via photosyntheses.

An animal is a truly multicellular organism with no cell wall. They are heterotrophic, meaning that they cannot produce their own food and must seek nutrients from sources.

A fungus is a truly multicellular organism with cell walls composed of chitin. They are heterotrophic.

Protists are typically unicellular organisms, though occasionally they may lump together to form something that looks like a multicellular organisms, though unlike the others they will have no tissues. Depending on the protist, they may be heterotropic or autotrophic.

Insects are multicellular organisms. The cells of an insect do not possess a cell wall. Insects obtain their nutrients from other sources by eating. This satisfies the criteria to be classified as animals.