Poe is considered the "Father of the modern-day short story." Why do you think his works remain popular more than 150 years later?

There are a number of reasons Poe's works remain popular today:
First, Poe came up with compelling plots. He knew how to tell a frightening story and how to build suspense. In "A Cask of Amontillado," for example, the reader anticipates that something unfortunate is going to happen to Fortunato, though Fortunato himself has no idea that a terrible fate awaits hims. Poe delivers a satisfyingly horrible punch when he has Montresor wall up the living Fortunato to die alone in a dark, damp catacomb.
Other stories play on subconscious archetypes. Poe's gothic sensibility allows readers to experience the dark, uncanny side of life that we often repress. Tales like "The Fall of the House of Usher," for example, allow us to explore eery archetypes, such as that of the doppleganger or twin, which is often a symbol of our own death. In German, the eerie, unsettling feeling that Poe's stories create is called "unheimlich" or "un-homelike." It would be hard to find a less homey house than Usher, and through it we can experience the nightmarish and irrational side of life and death. Poe, although predating Freud, has a firm grasp on psychology and lets us know loudly and clearly that there is much going on beneath our facades of sunny normalcy.
Poe had a talent for describing and imagining creepy environments: his stories are set in catacombs and dark, moldering mansions and involve unusual situations, such as threats of being sliced in half by a giant pendulum. In other words, they allow us to experience satisfyingly dark and gothic settings. He is a cinematic writer who is interested in creating an effect.
Poe not only wrote horror stories, but invented the modern detective story: he is popular because was a master at carefully plotting a mystery.
Also, unlike other writers of his time, he is not judgmental or moralistic: he leaves it to readers to decide for themselves what they think of his often unhinged characters. People tend to like to be allowed to experience their own emotions and draw their own conclusions.
Finally, we can't discount Poe's dramatic life, including his alcoholism and marriage to a thirteen-year-old, as well as his early death, in propelling the popularity of his stories. Also, many of his tales have been made into films—often repeatedly. According the Internet Database, more than 200 films are based on his work. Most people have heard of Poe, and knowing about him undoubtedly drives interest in reading his writing.