Ok, so Hotei is a Buddha(Just not the first one), and the reason why he is shown has Fat is to incorporate that he is full of love, compassion, wisdom, virtue, and so on.
There are many interpretations of Hotei I agree with the above but that is from the Chinese prospective and we call Hotei (Japanese Name) Budai or Putai in Chinese.
He to be an incarnation of Miroku Bosatsu known as Miluo Fo in China(or as Miluo Pusa) Known as the Fat Buddha or Laughing Buddha in the West.
Ok, I'll tell you everything I know about this topic as it seems to have peaked some interest.
In Japan Hotei is known as the god of contentment and happiness, and is the god of good luck from Shintosim, Hotei. He is supposedly based on Miluo Fo, and is widely recognized outside of Japan. He is generally depicted carrying a large cloth bag over his back, one that never empties, he uses it to feed the poor and needy.
The Japanese spelling of "Ho Tei" means "cloth bag." He also holds a Chinese fan called an
oogi (said to be a "wish giving" fan, this type of fan was used by the aristocracy to indicate to vassals that their requests would be granted. Hotei is most likely based on the itinerant 10th-century Chinese Buddhist monk and hermit Budaishi who is said to be an incarnation of Miroku Bodhisattva as I mentioned above
(Maitreya in Sanskrit for those interested).
Hotei is sometimes shown surrounded by a group of small children, playing and laughing around his very round shape.
In recent times, Hotei is also referred to as the patron saint of restaurateurs and bartenders I'm sure you see him in every Chinese restaurant in in the world (and on the dash of my car to remind me to be generous where ever I go). When you over eat or over drink (that would ever happen) the Chinese sometimes jokingly say your under Hotei's influence, well its funny in Chinese anyway .