This form could be easily confused with the previous 았/었더니 form, so the difference is that this form indicates that V2 occurred when V1 was occurring. It is used for cause-and-effect.
1. 계단을 내려가다가 넘어져서 팔을 다쳤어요. When I was coming down the stairs I fell and broke my leg.
2. 축구하다가 발목을 삐었어요. When I was playing soccer I sprained my ankle.
3. 요리하다가 손가락을 베었어요. When I was cooking I cut my finger.
Unlike in the 았/었더니 form, this form allows you to use actions that you could not control but that randomly occurred.
*위험하게 운전하다가 사고를 냈어요. When driving dangerously he got into a car accident.