How did Bilbo find Thorin? What message did Thorin want him to convey to other dwarves?

Bilbo snuck around the palace following the guards. He never took off his ring and rarely slept. He found all the other dwarves before Thorin and discovered Thorin's whereabouts by eavesdropping on the guards.

Eventually, after a week or two of this sneaking sort of life, by watching and following the guards and taking what chances he could, he managed to find out where each dwarf was kept. He found all their twelve cells in different parts of the palace, and after a time he got to know his way about very well. What was his surprise one day to overhear some of the guards talking and to learn that there was another dwarf in prison too, in a specially deep dark place. He guessed at once, of course, that that was Thorin; and after a while he found that his guess was right.

One day, to Bilbo's surprise, he heard the guard talking about another dwarf in a cell. Bilbo still had his ring put on. Instantly he guessed it would be Thorin. He followed the guards and found Thorin out. And his guess turned out real.

Thorin forbade all the other dwarves to tell the elf-king about their mission for the king will ask for a share of the treasure. Thorin was determined once more, after he lost hope, that he would not tell any of their goals to the king until the clever burgler, he himself or any other dwarves found a way out but if all the hopes of escaping had disappeared, they would think of letting the elves know.