밥상에 숟가락만 얹혔을 뿐이다. => 밥상에 숟가락만 얹었을 뿐이다. (lit) only put(얹다) one's spoon(숟가락) on the table(밥상) to eat (without taking part in preparing the meal). => (They / I) piggybacked on mostly other people's work. (= put in little effort and got undeserved rewards) 숟가락을 얹다 = a figurative expression for wanting to take one's share in something. 얹다 (root of 얹었다, its past tense form) = to put an object on top of something else (like 얹어 놓다). 얹히다 (root of 얹혔다, 얹히었다) = passive form of 얹다, but some people use it like a causative verb (히 sometimes makes both passive and causative verb, like 읽히다). So 숟가락만 얹다, 얹어 놓다, 얹히다 all seem to be in use, but 얹다 is probably the best one.

Translation wise, it means something like "someone who only puts the spoon on the table" As far as I can tell it has something to do with giving someone credit, so if someone only puts a spoon on the table he is responsible for the most part of an achievement, so someone who deserves the credit for that achievement. I'm not completely sure though