We consider trapped atomic Fermi gases with Feshbach-resonance enhanced interactions in pseudogap and superfluid temperatures. We calculate the spectrum of radio-frequency (or laser) excitations for transitions that transfer atoms out of the superfluid state. The spectrum displays the pairing gap and also the contribution of unpaired atoms, that is, in-gap excitations. The results support the conclusion that a superfluid, in which pairing is a manybody effect, was observed in recent experiments on radio-frequency spectroscopy of the pairing gap.