Why does Christopher Nolan end the Dark Knight Trilogy this way?

First of all it is an allusion to Alfred's previous story: When Bruce was away for many years during the events of Batman Begins (when he was an outlaw and later trained by the League of Shadows), Alfred was every years on vacation in that restaurant and hoped for seeing Bruce sitting there happily. He just hoped that he would finally have found a way to escape all the sorrow that Gotham meant for him.

As we learn in one of the ending scenes, the autopilot has been secretly repaired by Bruce a long time ago, which already suggests that there was something going on with the Bat. So Bruce really got out of it alive. And this is in line with the movie's earlier motive of Bruce rediscovering his will to live (think of the pit scene as a good example). After all his sorrows and secretly hoping to die for the sake of Gotham, he finally managed to find back to himself and discovered that there could be a Bruce Wayne without a Batman. He managed the "symbol" to bring the ultimate sacrifice for Gotham and remain in the minds of the people forever, while still not sacrificing the "person" and found a way to finally separate Batman from Bruce Wayne.

And this leaving behind of Gotham and all its sorrows is what Alfred always hoped for Bruce and in this last scene he (and the audience) finally found it. So what Christopher Nolan gave there was not only a happy ending for Alfred (and the audience) but the culmination of Bruce's struggle to get out of the pit he was in since the death of his parents (and to show that final conclusion for the main character is a strength only a completed trilogy can play out). After all, this path of Bruce to leave behind his sorrows and to finally cope with all the emotional trauma without just supressing it into the Batman was a major point of the the whole trilogy.

There might be a few people that are going to tell you that the ending scene was just a dream by Alfred and he just wished for Bruce to not have died, I for myself am pretty sure that Alfred indeed saw Bruce there at the end in reality, because of the above mentioned reasons. Still one could just see this whole 3-movie-struggle for peace of mind as unsuccessful and fruitless with Bruce's death as the only viable solution, but I refuse to do so for the reason that The Dark Knight Rises's whole motive of rediscovering his will to live suggests otherwise.

It's not "Alfred's rendition of what "the end" should have been". It's what actually happened. Everyone thinks that Bruce Wayne is dead and so is Batman.

That was Bruce Wayne's ploy. During the final 5 minutes it is revealed that "The Bat" had an auto pilot system. This proves that it is not necessary that Batman destroyed himself with the plane and the bomb. He could have easily escaped the plane leaving it on Auto Pilot, thus saving the city and himself too. This theory is verified when we see that Alfred sees Bruce Wayne and Selina Kyle in a cafe.

This is similar to the scene (imaginative), which Alfred had narrated to Bruce earlier in the movie - 'When Bruce was in China (Batman Begins) everyone thought that he was dead. When Alfred used to go to a restaurant/cafe, he always expected that one day Bruce would suddenly appear before him'.

In the end it just happens like that - "Everyone thinks that Bruce is dead but he suddenly appears before Alfred"

That's the significance of ending scene.

Bruce Wayne wanted somebody to take his mantle and protect the city. Robin was the best choice for him who gets the cave and all the gadgets to protect Gotham after Batman. Alfred dreamt about meeting Mr. Wayne with a girl at a cafe, they don't speak to each other but they just nod. This is the exactly the ending scene, like a dream come true. Films don't end at Climax, they must have conclusions. TDKR gave the epic conclusion to the trilogy.