"What is ___'s reason?" is a way of asking "why". Use this to ask why someone made a decision. For example:
What is their reason for wanting to sell it?
"What is ___'s reason?" sounds more formal than "Why". When you ask someone "Why?" instead in a formal situation, it might seem like you're criticizing their decision:
A: Why are you quitting?
B: (surprised) Well, I... I've simply decided that it's the right time.
"What's your reason for ___ing?" doesn't have the same emotional impact. This is a polite way to talk to a coworker who's leaving your company:
A: May I ask what your reason for quitting was?
B: Well, I guess I'm just burnt out and I feel like I need a change.