That's not true. Not only do fat cells just grow bigger (a skinny person has just as many as a fat person), you'd expect the increase in cancers to be mostly liposarcomas ("fat cell cancer"), which stay very rare.
Other organs might be distended or enlarged, but they don't contain more cells.
The reason obesity leads to cancer is not very well understood, but it's probably a complicated process involving abnormal hormone levels usual in obese people, and the constant low-grade inflammation their body is usually suffering from