-게 supplies a manner or degree adverb in front of 다치다 'get injured'. Severity words are common here — 심하게 다쳤어요 (I was badly hurt), 크게 다칠 수 있는 운동 (a sport where you can get seriously hurt) — but the lines also include wishes that nobody get hurt at all, as in 안전하게 다치지 않고 and 건강하게 다치지 마시고. 다치다 is always something that happens to you; to say you injured someone else you need 다치게 하다.