For my part the reason I love this couple so much is that they are shown to be a actually healthy couple throughout most of the series run even before they started to show signs that they were beginning to fall in love with each other. Unlike Korra and Mako Korra and Asami had built their relationship on foundation of friendship and had been there for each other at some of the hardest moments of each others lives. With Korra and Mako the two spent most of the first season arguing with one another and Mako repeatedly accused Korra of trying to ruin his relationship with Asami and use his brother to get back at him for not going out with her. And for Korra's part we have her treatment of him in the first half of Book 2 where she is just plain abusive towards him for no reason. Sure I don't like Mako but Korra was just being a jerk for no good reason.
Contrast this with Korra and Asami, the two do hit a few road bumps but they are able to work past the problem and talk it out and are able to see each others point of view on the matter. The only time they had a serious argument was in Reunion where Korra questioned if Asami's father could be trusted and Asami got angry with Korra on the matter. Korra realized that she hurt Asami's feelings and didn't escalate the argument and instead apologized for her poor choice of words.
To make a long story short, (to late), what I feel works best about Korrasami comes down to three things.
1: The two have great chemistry. Unlike either girls relationship with Mako Korra and Asami play off each other a lot more naturally with the two often being a lot more fun to watch.
2: Their relationship has a more natural build to it. With the first season Korra and Mako didn't really have a reason to fall for one another and, as stated before, spent most of the first two seasons arguing with each other. With Korra and Asami the two go from Aloof Rivals for Mako's affection, to friends, to partners in fighting crime, to best friends and finally to lovers over the course of 3 years out of universe and 4 years in universe.
3: They genuinely love each other. With Mako their interest in him was, IMHO, as superficial as you could get. Mako was created with the soul purpose of being Korra's love interest because he was hansom and brooded a lot. With Korra and Asami there was, again IMHO, always more to it then that. Korra at first saw Asami as nothing more then a prissy rich girl but came to see how much they had in common. Asami, on the other hand, had heard so much about Korra and over the course of the series came to be there for her at some of her lowest moments.
And that's why I think Korra and Asami are still such a popular ship even to this day. Why do you think they're such popular couple?
