"Moé" (萌え) literally means "budding", but it's Japanese nerd slang for the warm fuzzy feeling you get from certain anime and manga characters.
This is an escort service, but not an escort service. It claims to be companionship focused (as opposed to sex focused) for individuals who desperately want just that. A three hour date costs ¥13,550, while a five hour one costs ¥19,500. Those who participate in the service get their date's email address from the Moé Date folks.
Then after arranging a meeting place, the Moé Date cast member asks the guy where they'll be going on the date, which, according to Japanese site Rocket News, gives the feeling of a first date. All the pressure is on the guy to plan out the date. The Rocket News reporter took his Moé Date to a coffee shop and then to sing karaoke.
The Moé Date site is very clear to note that this is not a prostitution service, and it doesn't allow its staff to, say, go to a hotel for obvious reasons. There are a myriad of sexual services offered in Japan, and Moé Date isn't one of those businesses.
Recently, a "snoozing with women" establishment opened in Tokyo, and that service has turned out to be successful and is expanding to a second store. Perhaps Moé Date, with its focus on companionship, can fill that gap for lonely hearts during their waking hours.
萌えデート [Official Site]