A Carlsenesque position

My previous blog post featured my latest OTB game. I have not found time to analyze it until now, but now I have discovered why I could not find a win. I quite frankly do not understand the endgame. The conclusion is a bit painful, since it does not really give me anything specific that I can use in future games. If I could just spot some inaccuracy, I could learn from it and try to do better next time. But in this position, I do not know how to proceed, so the road to improvement is so much longer.

The position below shows the point in the game where I lost track of what to do. What is black's plan? 

With the bishop pair, a better pawn structure, and an extra pawn, this should be a win for black. And the computer agrees. In the comments I have included a line suggested by the computer at depth 25. Analyzing the position again after playing through the moves suggests that black has an increasing advantage.

I do not understand what to do here, even after having seen the moves suggested by the computer. In other words, I need to study the position more.