An old man, who happens to be endowed with a pair of enormous wings, crash-lands at Pelayo and Elisenda's house during a rainstorm. This is the setup for all the action that will happen afterward in the story. Since it's an unlikely (okay, completely impossible according to all known laws of biology) situation, it also tells us that the story is going to have a magical-realism bent.
Elisenda decides to start charging admission to see the old man, because there's a big debate in the community about whether or not he's an angel. (And it's also just kind of fun to poke him with sticks.) What's the problem? Well, did you read the bit about "poking with sticks"? The conflict comes from the way that something so magical and out-of-this-world is turned almost immediately into a money-making scheme that ignores and erases that beautiful, fantastic aspect of the old man.
As Elisenda and Pelayo count their money, the crowds mistreat the angel, poking, plucking, and pelting him with rocks. When they finally burn him with a branding iron the angel panics, scaring everyone with his reaction to the pain. We'll consider this the crisis, because it is the moment when the "angel" is getting the most attention, and making Elisenda and Pelayo the most money, and letting the community most blatantly shows its cruel side.
Just as quickly as it appeared, the crowd vanishes, distracted by the next big thing. When a carnival comes to town with a new sideshow, the masses disappear and leave the angel to just grow old in the yard of Elisenda and Pelayo's new mansion. Years go by without anything too exciting happening, and the old man's worsening condition lets us know the story is starting to wind down.
After a particularly hard winter, the old man takes a few practice runs then finally flaps his enormous wings and disappears into the horizon, much to the relief of Elisenda. Are you looking for answers? You won't find any in this sorry excuse for a resolution. (We kid, Gabriel, we kid!) Everyone is just relieved that he's finally gone.