Why didn't Thorin and the dwarves tell the king why they were in the wood?

Thorin didn't want the King to know they were searching for the treasure unless they had to..... in other words, they had no desire to share their treasure.

So it was that Bilbo was able to take secretly Thorin's message to each of the other imprisoned dwarves, telling them that Thorin their chief was also in prison close at hand, and that no one was to reveal their errand to the long, not yet, not before Thorin gave the word. For Thorin had taken heart again hearing how the hobbit had rescued his companions from the spiders, and was determined once more not to ransom himself with promises to the king of a share in the treasure, until all hope of escaping in any other way had disappeared; until in fact the remarkable Mr. Invisible Baggins (of whom he began to have a very high opinion indeed) had altogether failed to think of something clever.

The other dwarves quite agreed when they got the message. They all thought their own shares in the treasure (which they quite regarded as theirs, in spite of their plight and the still unconquered dragon) would suffer seriously if the Wood-elves claimed part of it, and they all trusted Bilbo.