-(으)려다 is a squeeze of -(으)려고 하다가, "was setting out to," and 보니까 or 보면 adds "and then I found." Together they explain how a result came about from an intention: [4인분을 하려다 보니 길어지는 준비], "trying to make four servings, prep drags on"; 거짓말을 하려다 보니까 이렇게 사레가 걸리네요, "that's what I get for trying to lie." Every line here is followed by the consequence. The close neighbour -다 보니까 reports the effect of actually doing something for a while; -려다 보니까 reports the effect of merely trying or intending to.