ackfoobar has already given a good reason why function types are called exponential, but there is an even deeper reason: function types interact algebraically the same way as exponents.
The type A → (B → C) is isomorphic to (A × B) → C (via currying). This is analogous to the rule (cᵇ)ᵃ = cᵇ˙ᵃ.
The type (A + B) → C is isomorphic to (A → C) × (B → C) (a function with a case expression can be replaced with a pair of functions). This is analogous to the rule cᵃ⁺ᵇ = cᵃ·cᵇ.
The type A → (B → C) is isomorphic to (A × B) → C (via currying). This is analogous to the rule (cᵇ)ᵃ = cᵇ˙ᵃ.
The type (A + B) → C is isomorphic to (A → C) × (B → C) (a function with a case expression can be replaced with a pair of functions). This is analogous to the rule cᵃ⁺ᵇ = cᵃ·cᵇ.