The truffle is a food item available for purchase at Food Mart. You cannot necessarily buy it on your first day of gameplay; it appears randomly. When you buy it for the first time, it will stay there, showing up as an "already purchased item". From the time of your first purchase of it onward, you may choose to buy it again at any time, provided that you have enough money to do so.
The surprising thing about the truffle in "Tomodachi Life" is that there are very few Miis who do not enjoy eating it; well, that, as well as the fact that any Mii who enjoys it even the slightest has a seventy five to eighty five percent chance of leveling up. The percentage varies depending on how much the Mii in question likes the treat. As far as has been tested, no other item in the game has such a high percentage of possibility of leveling any Mii up.