Oh for gods sake!
The research they are referring to is Keys et al, where he studied saturated fat intake in schizophrenic patients on a ward in the 1960's. He fed 60 inpatients a totally liquid control diet with varying levels of sat fat. The study showed that total cholesterol increased with increased sat fat diets, but that LDL lowered and it was HDL and triglycerides that increased and even that was marginal 
His methodology was massively flawed and the entire study made no mention at all of demographics like age of the patients, gender, weight, height etc and he also omitted 60% of his data that didnt fit his conclusions.
It makes me so so SO angry when bad science is not only used to support a point, but is then also misquoted! And the guys who said there's no evidence to support that sat fat isn't bad for you. There's actually 23 double blind, peer reviewed and published studies that i know of:
1.Foster GD, et al. A randomized trial of a low-carbohydrate diet for obesity. New England Journal of Medicine, 2003.
Details: 63 individuals were randomized to either a low-fat diet group, or a low-carb diet group. The low-fat group was calorie restricted. This study went on for 12 months.
Conclusion: There was more weight loss in the low-carb group, significant at 3 and 6 months, but not 12. The low-carb group had greater improvements in blood triglycerides and HDL, but other biomarkers were similar between groups.
- Samaha FF, et al. A low-carbohydrate as compared with a low-fat diet in severe obesity. New England Journal of Medicine, 2003.
Details: 132 individuals with severe obesity (mean BMI of 43) were randomized to either a low-fat or a low-carb diet. Many of the subjects had metabolic syndrome or type II diabetes. The low-fat dieters were calorie restricted. Study duration was 6 months.
Conclusion: The low-carb group lost significantly more weight (about 3 times as much). There was also a statistically significant difference in several biomarkers:
Triglycerides went down by 38 mg/dL in the LC group, compared to 7 mg/dL in the LF group.
Insulin sensitivity improved on LC, got slightly worse on LF.
Fasting blood glucose levels went down by 26 mg/dL in the LC group, only 5 mg/dL in the LF group.
Insulin levels went down by 27% in the LC group, but increased slightly in the LF group.
Overall, the low-carb diet had significantly more beneficial effects on weight and key biomarkers in this group of severely obese individuals.
You can find a bunch more here for low carb.
And here's a further 5 on sat fat specifically:
satfat
- Hooper L, et al. Reduction in saturated fat intake for cardiovascular disease. Cochrane Database Systematic Review, 2015.
Details: This is a systematic review and meta-analysis of randomized controlled trials, performed by the Cochrane collaboration – an independent organization of scientists.
It is probably the best review you can find on this at the moment, and includes 15 randomized controlled trials with over 59,000 participants.
Each of these studies had a control group, reduced saturated fat or replaced it with other types of fat, lasted for at least 24 months and looked at hard endpoints, such as heart attacks or death.
Results: The study found no statistically significant effects of reducing saturated fat, in regard to heart attacks, strokes or all-cause deaths.
Although reducing saturated fat had no effects, replacing some of it with polyunsaturated fat led to a 27% lower risk of cardiovascular events (but not death, heart attacks or strokes).
Conclusion: People who reduced their saturated fat intake were just as likely to die, or get heart attacks or strokes, compared to those who ate more saturated fat.
However, partially replacing saturated fat with polyunsaturated fat may reduce the risk of cardiovascular events (but not death, heart attacks or strokes).
These results are similar to a previous Cochrane review, done in 2011 (2).
- De Souza RJ, et al. Intake of saturated and trans unsaturated fatty acids and risk of all cause mortality, cardiovascular disease, and type 2 diabetes: systematic review and meta-analysis of observational studies. BMJ, 2015.
Details: This systematic review and meta-analysis reviewed observational studies on the association of saturated fat and heart disease, stroke, type 2 diabetes and death from cardiovascular disease.
The data included 73 studies, with 90,500–339,000 participants for each endpoint.
Results: Saturated fat intake was not linked with heart disease, stroke, type 2 diabetes or dying of any cause.
Conclusion: People who consumed more saturated fat were not more likely to experience heart disease, stroke, type 2 diabetes or death from any cause, compared to those who ate less saturated fat.
However, the results from the individual studies were very diverse, so it is hard to draw an exact conclusion from them.
The researchers rated the certainty of the association as “low,” emphasizing the need for more high-quality studies on the subject.
- Siri-Tarino PW, et al. Meta-analysis of prospective cohort studies evaluating the association of saturated fat with cardiovascular disease. American Journal of Clinical Nutrition, 2010.
Details: This meta-analysis reviewed evidence from observational studies on the link between dietary saturated fat and risk of heart disease and stroke.
The studies included a total of 347,747 participants, who were followed for 5–23 years.
Results: During follow up, about 3% of participants (11,006 people) developed heart disease or stroke.
Saturated fat intake was not linked with an increased risk of cardiovascular disease, heart attacks or strokes, even among those with the highest intake.
Conclusion: This study did not find any association between saturated fat intake and cardiovascular disease.
Keys is literally the only researcher in history who has skewed his data sufficiently to show satfat has an affect on heart disease risk factors, and the guy was a quack. Somehow though he stirred up such a revolution that we based 40 years of nutritional advice on his awful, awful science 
Sorry, i went off on one a bit. I'm a total science geek so bad science really irks.