Attached is a portion of my stats textbook.

Can someone explain how the approximation at the bottom was obtained?

I believe this is the "delta method". It is derived from Taylor expansion:

mu(T) = mu(theta) + (T-theta)*mu'(theta) (approx.)

so that:

Var[mu(T)] = Var(T)*[mu'(theta)]^2 (approx.)

theta and mu(theta) are constants.