I don't think he pronounces a glottal stop. He certainly doesn't pronounce a final /d/, and I'm not even convinced he uses a consonant at the beginning. Turn the volume right up, slow playback down to 25%, and you'll hear it much more clearly.
Remember that you don't need to hear every sound of a word in order to understand it. The fact that you didn't understand what he said was not so much a result of you not hearing any particular words rather than you being unfamiliar with the phrase 'That would be sick'.