Ahhhhhh, Dain is a fun character to write – because he is so blunt and honest and funny, but he is not necessarily forthcoming. He is very, VERY smart – easily one of the smartest characters in the whole story. He is also very perceptive. He can conceal all of this behind his loud, bluff, good-humoured exterior. People don’t expect that savagely sharp brain: they expect a Dwarf like that to be dumb. 

Last mistake they’ll ever make, lol.

Dain knows his cousin. He knows that Thorin tends to latch onto causes, and he knows that they always have huge personal significance to him.

So, Thorin knew a hobbit – and there would have been rumours about that floating around the remainder of the Company the whole time Dain ruled Erebor. Nothing explicitly said, just hints here and there, but Dain is easily intelligent enough to pick up on nuance. He’d know that there was  something unsaid, something there, between Bilbo and Thorin. 

Therefore, Dain would know that Thorin’s interest in these two hobbits must have personal significance as well. It’s not too hard to put two and two together to make four: particularly after discovering that Merry and Pip are Bilbo’s cousins.