Can someone explain to me in a brief sentence what this excerpt means? The “new addition” which Hume goes on to provide in “Of Tragedy” is based on the thought that the two affective elements in our experience of tragedy have different causes. On the one hand, the “sorrow, terror, anxiety, and other passions, that are in themselves disagreeable and uneasy,” are generated in us by our sympathy with the characters represented in the work. The pleasure that we experience, on the other hand, is initially and primarily a result of our attention to “that very eloquence, with which the melancholy scene is represented.” The display of “genius,” “art” and “judgment” that we see in a “well-written” tragedy, “together with the force of expression, and beauty of oratorial numbers, diffuse[s] the highest satisfaction on the audience, and excite[s] the most delightful movements.” This satisfaction and delight, being, as Hume has it, the dominant element of the experience, “overpowers” and somehow “converts” the negative feelings: The impulse or vehemence, arising from sorrow, compassion, indignation, receives a new direction from the sentiments of beauty. The latter, being the predominant emotion, seize the whole mind, and convert the former into themselves. . . . As a result of this process of conversion, “the soul, being, at the same time, rouzed by passion, and charmed by eloquence, feels on the whole a strong movement, which is altogether delightful.” (OT 219–220)

I can't summarize this excerpt in one brief sentence, but I shall do my best to summarize it for you as succinctly as I can.
The excerpt is about an essay, entitled "Of Tragedy," published in 1757 and written by David Hume, a Scottish Enlightenment philosopher. In the essay, Hume tries to make sense of why people enjoy watching tragedies. He points out the peculiarity of people being "pleased in proportion as they are afflicted" and of being happy to "employ tears, sobs, and cries to give vent to their sorrow."
In the first half of the excerpt you have provided, the author gives a summary of the problem with which Hume begins his essay, which is namely the problem of how people can enjoy experiencing, by watching tragedies, "disagreeable and uneasy" emotions. The "disagreeable and uneasy" emotions are an expression of our sympathy with the characters who we are watching endure tragic circumstances. The "pleasure that we experience," on the other hand, is a result of our appreciation of the craft with which those "disagreeable and uneasy" emotions are communicated by the actors and the writers.
In his essay, Hume proposes that we enjoy watching tragedies because we enjoy the eloquence of language and the craft of the actor that is needed to communicate those profound, difficult emotions. Tragedy is, Hume suggests, the form which best lends itself to beautifully crafted language and beautifully crafted acting. And this is because tragedies deal with the most complex and profound emotions, which only the most skilled craft can communicate.
In the second half of the excerpt, the author says that, according to Hume, the pleasure we take in the craft necessary to communicate tragedy "somehow" overpowers or redirects the negative feelings that we sympathize with. In other words, our appreciation of the necessary craft is stronger than the sadness we may feel or sympathize with in the characters. This is a reasonably accurate, fair representation of Hume's views, although Hume actually goes a little further.
In his essay, Hume does indeed argue that the negative feelings we sympathize with in a tragedy are overpowered by our positive appreciation of the craft. He also argues, however, that the stronger the negative feelings we sympathize with are, the more beautiful the craft will need to be, and, therefore, the more positive our appreciation of the craft will also be. Hume uses the analogy of a parent's feelings for a sick child. The sicker the child, he says, the more the parents will love that child. To directly quote Hume: "Parents commonly love that child most, whose sickly infirm frame of body has occasioned them the greatest pains."
I hope I've managed to explain clearly for you the meaning of the excerpt you provided. The excerpt essentially summarizes what is a reasonably brief essay by David Hume, although to my mind the summary is not as clear as the essay itself. I have provided a link to the essay below.
http://www.english.upenn.edu/~mgamer/Etexts/hume.tragedy