Or then that he does not, and the Klingon is being an asshole. Which is sort of the one and only dramatic point of that exchange.
Timo Saloniemi
To be fair, the translators are there for a reason, even if the film can't decide of they are universal translators or radios to the anachronistic UN style room of Kirk's drinking buddies from the last film. The film also very strongly implies that though we are hearing English, they are speaking Klingon, especially if there's an English accent involved, hence that translation line....it's either that, or the judge, prosecutor and defense, and the one Klingon with a sense of humour in the court are all speaking English, and the translators are purely decorative; with the Klingons so beaurocratic in their Soviet communist way that they have a room of people translating thongs that are either already translated by machines or being spoken in English. The film even almost makes fun of this by having the 'original Klingon' line about Shakespeare (plummer, Warner and Shatner are all members of Shakespeare companies, Shatner and Plummer being in the same one in their past.)
I would also disagree with Kirk repeating Kruge verbatim, and definitely get the sense that he spoke Klingon, and while it may have been unusual for American Captains in the Cold War to speak Russian (after all, those naval intelligence chaps had to have some use) the era for inspiration particularly in Meyers films is more the British/French conflicts of Hornblower etc. In that time of course, officers, particularly Captains, would speak and read French just fine, and vice versa.
Even in the second world war, higher ranking British Naval officers would of course be able to speak and read German, you need to deal with intercepts, prisoners, deal with the possibility of speaking for yourself or your men in the advent of capture...and let's not forget Kirk was an admiral for a good number of years.
Naval warfare was a different beast in the past, and Trek takes its cues from those, and implies it will be again in the future.
It's the same with the 'officers English' on the Klingon ship. Crewmembers trained as officers learn the enemies language as part if their training, and use it amongst themselves.
There is no 'dramatic point' to that individual line or delivery beyond the historical allusion, the scene itself is the drama of course, with Kirk being led to entrap himself via the responsibility of a captain for the crew under his command. Now Bones on the other hand should theoretically have walked free, even under Klingon law, unless not performing the Klingon Deaf...I mean Death...Wail is punishable by...erm...death. That or straddling the Chancellor is considered a great faux pas, and he should have at least nibbled Gorkons cheek first.
The scene was daft and daffy, I too like the previous posters suggestion concerning Uhura and Chekov's unused book. (for extra textual humour, we could have seen him reading it in preparation for the mission. Chekov's Hardback therefore becomes the antithesis for Chekov's Gun)
That's the thing about St6. It doesn't stand up very well to picking at it, less so even than V, even if it was much more fun for all of us at the time, and is something of a sacred cow for various reasons. This is especially true of various tiny scenes that the plot hinges upon (Spock's huge tracking device that is in no way disguised and dependent on Kirk keeping his coat, Romulan Ambassadors in top secret missions, extremely complicated conspiracies to achieve simple ends, members of the crew being handed the idiot stick even more than in V...Though usually to make Spock look good rather than Kirk this time. Or, as in the courtroom and with 'i don't even know the anatomy' simply to up the drama in tense scenes.)
But hey...it's what fans do. In this case, it's also why I am fifty fifty on Meyer being involved in the new series should be seen as a promising thing.