First, it brought back stability. Nolan co-wrote and directed all three movies, so the tone, visuals, and feel remained the same. Just as importantly, Christian Bale was in the lead role for all three. Regardless of whether you think he was the best Batman/Bruce Wayne or not, the continuity that not changing actors every movie brought was very important. Secondly, the movies were more than just superhero or action movies. It brought in big-name actors for both the main roles and the supporting roles and focused on not only having good scripts but excellent acting (punctuated by Heath Ledger's performance as The Joker). Those elements were spotty in the superhero genre up to that point. In some movies, you would have big-name actors and a good script. In some, you would have a decent script but only one or two big-name actors, and in some, you would have an awful script. These movies did not rely just on a bunch of special effects and lots of explosions to tell the story. Lastly, Nolan tried to bring a sense of realism where possible. Obviously, these are movies based on a comic book, and most of what you see in them could not happen in real life. But, the fact that the movies were filmed in real cities and that the gadgets he used (especially in Batman Begins) were somewhat realistic (like his grappling gun) made the movies more enjoyable for me.
Of the three movies I actually think the last one has the best overall story. Obviously, however, Heath Ledger's performance as Joker was the touchstone of the three movies. He played the in-control psychotic so well, that it will be hard for anyone to top that live-action version of the character. Even though I think Ledger was the best villain, I think all the movies did a good job with the bad guys. I especially liked that they cast Bane based on acting ability and not just finding a big guy with no talent. Obviously, Tom Hardy is not 7 feet tall or anywhere near it, but his version of Bane was much better than the one in Batman and Robin.
Chances are if you are reading this you have already seen each movie multiple times. While I do not think any of them are absolutely perfect, I think they are as close to perfect as the superhero movies that had come out up to that point had been in a long time. And, you can certainly argue that the success of Batman Begins played a big part in the Marvel movies being taken seriously from the start. I had hoped that the series that Nolan started would continue in some way. The ending of Dark Knight Rises certainly left that possibility open. But, after this many years, if it were to do so, it would more likely be in a "Batman Beyond" type of situation in which Bale plays the role of Alfred to a new, younger, Batman.