I've always liked this exchange. Today, they (or rather their lawyers) would be doing this in a courtroom.
Bard makes two claims. The first is personal, divided into two parts. The first part is that he is the slayer of Smaug, the implication being that he deserves a reward for this. The second part is that he is the heir of Girion, Lord of Dale, whose wealth is part of Smaug's hoard. The second claim is on behalf of the Lake-men, he still being the servant of their master. He asks if there is any thought for their 'sorrow and misery', they having earlier aided Thorin and his people.
Thorin's response is that Bard put his worst cause last and in the chief place. He said that no man had a claim on the treasure due to the destruction of Smaug; because the treasure wasn't Smaug's to begin with. While they would 'fairly’ pay for the goods and assistance the Lake-men gave them, they would give nothing under threat of force, while an armed host 'lies before our doors'. He then asks the pointed question about how much of the treasure Bard and his allies would have given to their relatives, if they had found it unguarded and he and his people killed.
Bard sidesteps this, saying that they were not dead, and he and his allies 'not robbers'. He then said that the wealthy may have 'pity beyond right' to help the needy who befriended them when they were in want. Also, his other claims remained 'unanswered'.
Thorin replies that he would not negotiate with armed men at his gate, still less with the people of the Elvenking 'whom I remember with little kindness'. He then tells Bard to go 'ere our arrows fly!'
From a legal point of view, Bard has a valid personal claim. As slayer of Smaug he would be entitled to a reward from those who benefited the most from that dragon's death. Also, as the heir of Girion, he would have a valid hereditary claim to the relevant parts of the treasure that were his ancestor's. It's interesting that Thorin refuses to discuss this claim, admitting its great strength.
In terms of whether some treasure should be given to the Lake-men, Thorin concedes that he and his people will pay fairly for the goods and assistance given them; but he refuses to agree any other legal liability, pointing out quite correctly that Smaug had no right to the treasure. For that reason, the Lake-men could not legally claim any part of it as compensation for Smaug's destruction.
But while Thorin is legally correct, Bard points out that the Lake-men helped Thorin and his people when they were in want, without asking for anything in return, appealing to their pity. This is a good point; and it would not only be the morally right thing to do, but also good politics, ensuring the goodwill of many towards the re-established Kingdom under the Mountain.
My own view is that Tolkien is showing that Thorin is morally, as well as mostly legally, in the wrong. It would be the morally (as well as politically) correct decision to give something to help the Lake-men, who had helped him and his people without asking anything in return. However, Tolkien gives as a reason the influence of the dragon-infected gold on Thorin, as well as the aggravating factor of the presence of an army on his doorstep, partly made up of Woodelves. There is the maxim 'The enemy of my enemy is my friend'. One could here twist it around to mean 'The friend of my enemy is my enemy', viewing the Lake-men in a hostile light for being friends with those elves who had imprisoned him.
Thorin is wrong, but Tolkien gives us reasons for understanding why. What do other people think?