Sir Gawain and the Green Knight begins with a short recap of the Trojan War because the author desires to put the grand reign of King Arthur in historical context. Within thirty short lines, the author gives a very poignant summation of English history. By beginning with "Since the siege and the assault was ceased at Troy," the author proves that he is mentioning only the most important of events and eventually filtering down to King Arthur. Ironically, however, it is a few lines down (from lines 20-26) that contains the support you are craving for the reason.
And since this Britain was built by this baron great, / Bold boys bred there, in broils delighting, / That did in their day many a deed most dire. / More marvels have happened in this merry land / Than in any other I know, since that olden time, / But of those that here built, of British kings, / King Arthur was counted most courteous of all. (1.20-26)
In other words, Britain was built by great people and many amazing deeds have been accomplished there, but of all the marvels that have happened, the reign of King Arthur and his court is counted as the best of all. Therefore, with all of the historical studies that we adhere to about Troy and Rome and France, nothing can top the story of England's King Arthur.
In addition, it is important to say that the poem not only begins with Troy but also ends with Troy, framing the story quite nicely. Therefore there is another vastly significant quote from the very end of the poem:
After the siege ceased at Troy and the city fared amiss. / Many such, ere we were born, / Have befallen here, ere this. / May He that was crowned with thorn / Bring all men to His bliss! Amen. (4.2525-2530)
Again the reader can see the author's point: to glorify the reign of King Arthur and, even further, to show that under King Arthur England glorifies Jesus Christ.