His point is that if the integral defining $f(t)$ converges for all real $t$ then $f(t)$ on the real axis is the boundary value of a function that is analytic in the lower half plane. (Observe that taking $t\to t-i\tau$, $\tau>0$ improves the convergence of the integral) Now analytic functions that have a limit point of zeros (as is guaranteed by their vanishing in a finite interval) in the interior of their domain of analyticity have to vanish everywhere in that domain. What is not immediately clear to me to what extent this is true for functions vanishing the boundary of their domain. I suggest that you look up "Hardy Space" on Wikipedia.

His point is that if the integral defining $f(t)$ converges for all real $t$ then $f(t)$ on the real axis is the boundary value of a function that is analytic in the lower half plane. (Observe that taking $t\to t-i\tau$, $\tau>0$ improves the convergence of the integral) Now analytic functions that have a limit point of zeros (as is guaranteed by their vanishing in a finite interval) in the interior of their domain of analyticity have to vanish everywhere in that domain. What is not immediately clear to me to what extent this is true for functions vanishing the boundary of their domain as the boundary limits can be quite singular. For such limits, I suggest that you look up "Hardy Space" on Wikipedia.