-다가 means "in the middle of doing X," and 지치다 is running out of energy or patience — so the activity itself wears you down. Laughing is the commonest cause here: [웃다가 지친 나영 발견], [BH 형 때문에 웃다가 지침]. It also covers thinking (항상 생각하다가 지쳐서 포기해요), hearing the same thing over and over (그 말을 듣다 듣다가 지쳐서 그만뒀어요), and searching (그 뿌리를 찾다가 지쳐요). Note it often shades into "sick of it" rather than merely tired.