As others have indicated, the statement means that he completed his side of the bargain exactly as detailed. However, following something to the letter often includes the implication that you have violated the spirit of the agreement while conforming to the exact requirements. Vladimir makes this statement to Yeuh because he has also conformed to the letter of the bargain. Specifically, he has delivered Yeuh's wife from agony - by killing her. If she is dead, she is not in agony and so the Baron has fulfilled the letter of the bargain. Generally, it would be assumed that delivering her from agony would mean to release her from captivity and torture. (Yeuh makes it clear that he was not fooled by the Baron's duplicity and fully expected the Baron to kill her.)
It is common to see this usage of the phrase when saying that something conforms to the letter of the law.