Plot in The Importance of Being Earnest
Recall that in Act I Jack asked this very same question of Gwendolen, who took the same position as Cecily. Wilde uses this repeating plot line to draw direct parallels between the two main characters, Jack and Algernon, as well as their significant others. In this way, we see that Jack and Algernon, though markedly different in both their moral code and sense of humor, respond the same way to this situation, making them, if not brothers by blood, brothers in spirit.
Keep in mind that, as Lady Bracknell's nephew and son to her sister, "Ernest" is technically Gwendolen's first cousin, and it's improper for them to marry, in spite of the British aristocracy's tendency toward such incestuous relationships. It's likely that the lovers will overlook their familial ties and proceed with their engagement, but it should be noted that this isn't the clean, easy ending that they believe it to be.
At last we learn the real reason for Jack's insistence that he "[has] no brother, that [he] never had a brother, and that [he doesn't] intend to have a brother." His somewhat over the top protests set the stage for this comedic twist, in which it's revealed that he's actually Algernon's older brother. Now we understand that Wilde was hammering home the point about him not having a brother to prepare us for this moment.