-다 싶다 means "it feels like, as if maybe; think to oneself that"
it is used after a plain style quote.
피곤하다 싶으면 영락없이 도진다.
Whenever I feel tired, it hurts like hell.
지난 번이 마지막이다 싶었는데 또 친구에게 아쉬운 소리를 해야 한다.
Although I felt that was the last time, I must ask a favor to my friend again.
한국 사람들은 사적인 일에 관심이 많다 싶어요.
I get the feeling that Koreans have much interested in private stuffs.