가 in this case is not the subject particle. It means getting/going(meaning gradually) So it means "As I am getting older and older, I am understanding the world more and more."

The 가 at the end of both verbs comes from the verb 가다, which I presume you already know. So in this case it's putting emphasis on the movement from 'not knowing' to 'knowing'. In English you could translate 세상을 알아간다 to 'come/get to know the world'. 나이를 막어간다 is similar.. it just emphasises the fact that ageing is a process and takes time.

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