보자 is the plain 'let's look' form of 보다, used exactly the way English uses 'let's see…' while you check something or stall for a second: 어떤 질문이 있는지 보자 보자 'let's see what questions there are'. Repeating it is filler while you look. Quite separately, 보자 보자 하니까 is a fixed idiom — 'I've been letting this slide, but now…' — a warning, real or mock, that your patience has run out; two lines here use it that way.