Technically it means that the responder is just asking verification that it is going to happen, but it more implies that the person might be slightly surprised by the statement. Example:
"I'm going to be a father."
"Are you really?"
Sometimes it can be just a way to let the speaker you acknowledge you're listening though. In the sentence you provided: "I am thinking of starting a website." it may appear as if they are hinting at wanting a response of somesort. Someone who doesn't have an immediate opinion might say "Are you really?" simply to que the speaker to give more detail.