뚫다 is making a hole through something or forcing a way through: getting your ears pierced, drilling with your fist, a road cut through for residents, squeezing through a narrow gap. Half these lines ask 어떻게 뚫어 — how do you get through that. The -게 slot gives the manner word, such as 깊게 (deeply) or 시원하게 (clearing it right out).