Why does Prince Prospero hide in his palace in "The Masque of Red Death? How does life contrast outside the place with life inside it?

A clue to the answer to this question can be found at the very start of this short story. The "Red Death", a hideous pestilence that causes bleeding, red stains, pain, and "dissolution" in its victims, has reached the Prince's country and is ravaging the inhabitants away. 

But the Prince Prospero was happy and dauntless and sagacious.

Here is the first important clue for your answer. Prince Prospero is someone who is aloof from reality and disengaged from his people. Therefore, he continues with life, as he knows it. In light of the devastation taking place, he decides to call up his friends from court and hide away from the disease in one of his abbeys. 

Prospero thought that the iron gates of the abbey, the provisions, and all of the precautions that he could possibly take were a way to avoid contagion. Moreover, since he was used to a plush and lofty life, he chose to make this a jolly occasion, complete with a masquerade to keep himself and the courtiers entertained. 

The prince had provided all the appliances of pleasure. There were buffoons, there were improvisatori, there were ballet-dancers, there were musicians, there was Beauty, there was wine. All these and security were within.

In high contrast as to what was going on inside of the abbey, things were getting worse in the city. The pestilence had already been around for nearly six months, and was becoming worse than ever. 

Back inside, however, Prince Prospero had lavishly decorated the seven chambers of the abbey, and his eccentric and expensive tastes were evident in all of the sumptuous details that he considered. These included the large ebony clock, the drapes, costumes, and especially, the color of each room.

The lesson learned is that fate cannot be challenged, nor changed. Money and riches cannot be used to control life or death. Prospero used his power to escape from something inescapable. He thought that, by hiding from reality within the walls of a decorated abbey, he could escape it altogether. We know that, in the end, the Red Death comes for Prospero anyways. He will be a victim no matter what he tries to do.