Fat does not break down into sugar. It is either stored (in fat as fat) or it is oxidized (burned).
Many people do not realize that fat does get stored even if your in fat burning mode and even when your not eating carbs. If someone in ketosis sat down and ate or drank 200 grams of fat in one meal, (1800 cals) that fat will go into storage, but it will come out as needed for fuel.
You are also right, that in a mixed meal of course, fat gets stored then too. The difference is in a mixed meal with carbs, fat burning is turned down low in the presence of insulin. This allows the glucose to be burned off for a time period directly related to how much glucose is present.
At a high level, in terms of the bodybuilding diet with high carb+protein, he's correct. You eat that high carb/protein + high fat and the fat gets stored.
In detail however, he's not correct. Yeah, fat can get converted to some glucose, but it's a fairly negligible level and as needed, not all the time. It's not a major threat. You eat low carb/high fat and the majority doesn't get stored, it gets used, etc.