By using "가지고", you can describe "찌개를 끓였다" as a result of "고기를 잡아". Then, you can infer that the speaker used "고기" as an ingredient of "찌개". If you write "고기를 잡고 찌개를 끓였다", then there can be no causality, because "-고" is used in many ways - not only showing causality, but also simply "and", so you may feel that it is ambiguous.
Other examples using "가지고":
- 날씨가 너무 더워 가지고 공부를 못하겠다. (It is too hot, so I can't study well.)
- 그렇게 놀아 가지고 시험에 붙겠니? (You play too much, can you pass the exam? - indirectly blaming "You can't pass the exam if you play too much.")
Examples that "가지고" and "-고" are different:
- 여름에는 비가 내리고 겨울에는 눈이 내린다. (It rains in summer and it snows in winter.)
- 여름에는 비가 내려 가지고 겨울에는 눈이 내린다. (It rains in summer, so it snows in winter.(?))
Examples are from Naver Dictionary"가지고" and "-고" from Naver Dictionary.