High fat doesn't have to mean swimming in oil, this could mean for 2 fried eggs cooked in a teaspoon of coconut oil, or drizzling 2 teaspoons of virgin olive oil on a salad. As carbs are restricted on LCHF, the amount of fat in proportion is larger than standard, just due to the physical amounts. High fat is also about options such as selecting fattier cuts of meat or eating more oily fish, and eating nuts. Some say the "H" in LCHF can stand for "healthy" rather than "high".
My version of LCHF is mainly higher Keto based, so anything around 50 grams of carbs per day, sometimes a lot less. The saturated fats I get are mainly from meat, fish, nuts, seeds, butter, some cheeses, dark chocolate, coconut / mct, avocado, olive oil's and non-grain based flours.