Note: this got a bit long, so the TL:DR is, the best diet is one that fits into your lifestyle and helps you maintain a caloric deficit.
In simple terms, your body weight is a balancing act between calories in vs calories out (calories in being what you eat / drink, calories out being what you burn through exercise, everyday life, or pass naturally from your system), arguably your body composition is a symptom of the quality of the food you eat, but that's a completely different topic.
In order to lose weight, you have to shift the calorie equation more to the right (i.e. your calories out are greater than your calories in), to gain weight, you have to go the other way. There's no way around this, it's just how we understand* the body to work.
Understanding this, the answer to your question
Does the low carb Atkins type diet really works when it comes to losing fat?
is yes, as long as you maintain a caloric deficit.
To address your primary question
Is LCHF diet the best and easy way to loose fat?
The answer is, it depends.
A lot of people read about the latest diet fab, low carb / high fat, high carb / low fat, vegetarian, vegan, weight-watchers, zone diet, gluten free, slimfast plan, and immediately assume it's going to magically help them lose weight, they hit it hard for a week or two, then they relapse into their old habits because they find that being vegan is really hard when your family invites you to a BBQ on a sunny weekend, or low carb / high fat or gluten free is really tough when someone brings a box of doughnuts into the office, or high carb / low fat is very confusing when people tell you salmon and avocado are healthy foods (but they're so high in fat!).
The best diet is the one can you stick to.
I want you to read that again. Then think about your life, your day to day existence and really consider, I mean actually seriously consider if you'll be able to stick to it.
Lets say you're going to go with an Atkins style low carb / high fat diet (I haven't read the Atkins diet, but hopefully this isn't too far off the mark), consider the usual daily meals
Breakfast - You can't eat cereal, toast, crumpets, pop-tarts, or pretty much anything that comes in a box. A lot of people have eggs for breakfast (a good choice by the way), but eggs aren't really a grab and go food unless you remembered to boil them the night before, which means you now have to get up and make breakfast before work. What about if you've got kids and you have to get them ready for school, do you have time to cook for yourself? What if your alarm doesn't go off, you oversleep and have to rush to work?
Lunch - You can't nip out and grab a sandwich, or jacket potato, or sausage roll, or anything with rice or pasta. Which means either a salad, a restaurant where you can order steak and veg (no chips now!), or something pre-made that you've brought in with you (gotta love those business meeting sandwich buffets, where you're left with carrot sticks).
Dinner - This one is, usually, easier at least, as most people have time to cook, but this is also where you find out that eating out with friends / family can be tricky, especially if your partner has a love of Italian, Chinese or Indian food.
Desert - Ha. Ha ha. Hahahahahahaha. Wait, you're serious? Ok... there are a number of deserts you can make yourself, and you may be ok with some yogurts, but most deserts aren't that Atkins friendly.
I eat a gluten free diet, which I believe helps me maintain a relatively healthy physique (that's along with regular weight training, and despite working a desk job), so I tend to veer towards a low carb / high fat diet, so a lot of the above is from personal experience.
My meals will generally look something like:
Breakfast - fish or scrambled eggs, avocado, sauté veg and a smoothy (made from more vegetables plus a banana and a little protein powder). Sometimes I'll treatmyself to a bowl of gluten free granola, and I do keep some in the cupboard for emergencies (that thing about your alarm not going off? Yeah, that counts as an emergency).
Lunch - leftovers from the night before, generally a stew type concoction (gotta love that slow cooker) which is usually just meat and veg, or a frittata. If, for some reason I don't bring lunch with me, then it's a tin of fish with rice and sweetcorn (yes, rice is high carb, so you'd have to swap it out for salad).
Dinner - Remember when I said about loving the slow cooker? Yep, it's generally a slow cooker stew type things that I've prepared either the night before or that morning; or some form of meat and veg, sometimes with rice if I've trained that evening.
Desert - No.
If you don't think you live like this, and there's honestly nothing wrong with it if you can't (I don't eat a gluten free diet by choice, I eat it because I have to), then no, a LCHF diet isn't the best and easiest way to lose fat for you.
*at the moment, anyway, until we realise that everything is actually controlled by the Flying Spaghetti Monster