Katakuri’s past is explained in detail in Vivre Card Databook – Vol.12.
Katakuri was born on a ship 48 years ago alongside Daifuku and Oven, likely due to Big Mom’s travels as a pirate in her early life (we even see Katakuri on a ship in his flashback). At the age of two, 46 years ago, Katakuri tasted donuts and fell in love with them, becoming his favorite food.
However, just a year later, 45 years ago, Katakuri had a grave accident: in a frenzy of devouring donuts, he stuffed so many in his mouth that his cheeks literally ripped apart, leaving a massive gash at both sides of his mouth. Despite this leaving a deep scar, Katakuri contuinued to eat donuts day after day, without caring what people thought of him for loving donuts so much.
At the age of 5, 43 years ago, Katakuri ate the Mochi Mochi no Mi, obtaining its powers. Unfortunately, something happened another 5 years later, when Katakuri was 10.
His little sister Brulee, who was five at the time, was scarred by some thugs that wanted to intimidate Katakuri. Because of that, Katakuri decided to permanently hide his mouth and scar from others, keeping up the appearance of being a perfect man.