Hiya
You need to get your daily calories from somewhere. If you eat fewer than your base metabolic rate your metabolism will slow right down and you will find it impossible to lose weight.
Fat is a good source of dietary calories. Your other choices are protein and carbs. You know the result of eating carbs... I think it would be hard to get the calories you need from protein. Also, protein converts to glucose, and glucose needs insulin. Insulin stores fat. If you're running on fat and ketones, your body will not be producing very much insulin. So no fat storage. You'll be producing glucagon instead. Glucagon stimulates fat cells to release triglycerides into the bloodstream...

Hope this makes a bit of sense.
S