Well, this is a misleading title and unfortunately you can find it everywhere. It is only partially true. You cannot actually exercise yourself out of a bad diet ! However, fitness is part of health and can aid somewhat weight-loss.
So yes, actually your diet should be supplying you with leanness and health. Sports should be providing you fitness and shredded body. There are so many super lean people doing zero sports, just light activities like walking to the shop and collecting flowers. There are also tons of over-fat sportsman.

Burning calories, especially fat calories can accelerate weight-loss. If your diet and nutrition periodisation is correct, exercise, together with quality eating habits is your best bet for loosing that extra 100 or more pounds. We don't talk about the diet part here. It can be healthy eating, high carb vegan, 80/10/10, zone, low carb paleo, time restricted feeding, vegetarian and so. It is your choice. My advice is eat fruits & vegetables. The rest is compromise. There is good and bad compromise. Grains like rice, flax seed, some nuts, some algae, some little fermented soy, some pulses like beans, garbanzo, green peas can be part time to time. Meat, fish and eggs, can be part on a weekly or monthly base. This will be your decision. Processed foods are not really part of a humans diet. Just think about it. Coconut oil, palm oil, olive oil, almond milk, soy milk, honey, maple syrup, dried fruits and so. These are great I would say sports products. Why ? Cause they digest easily, can add calories without overloading the system and a very easy way to spend money on stuff too. However, when weight loss is a goal, avoid those ! The worst is highly processed complex foods. Sausages, pink ham, preserved bacon rolls, pre made biscuits even if it was organic, triangle sandwiches and all restaurant food.
Either ways, diet should be personalised and periodised. Whatever goes to you from fruits, veggies, meat fish and eggs.

Training

Running is fantastic. Fantastic as a fat burning aid. However there is one slight problem with it. Impact. If you had 200pounds extra, or even if it was only 20kg, you don't want that pounding. It will kill your joints. Including all of them ! However if running is your goal, you can include run technique training to your regime. You can go to the local track and see how the guys workout. Learn some drills and you can imitate it at home or down in the local park. If that was your first time, don't rush it ! 15 to 20minutes will already tire you from these schooling drills.
When your body will be ready, what is maybe in 3months, maybe in 13months, you'll have the technique and can run !

It is more natural to walk, than to run. Especially walking fast, walking uphill, walking with a backpack. No impact. Just make sure you focus on technique and breathing. You can wear an HR monitor and focus on the lowest heart rate possible. The MAF method is great for this. 180 - your age - 10 (- 10). This means that if you were 30 years old, your heart rate range will be 130 to 140bpm for fast 1 to 2hours of training. Or lower !
Guess what. Walking is very natural. If your goal is to really loose weight, you can walk all day long, after a while.

Second to walking is cycling. I think it is even better actually than walking, but financially it is less attainable to some. You need a bike and you must maintain your bike. Knowledge of components and cycling road rules are crucial. You must know how to fix a flat, a rubbing brake and also must know what is your correct position on the MTB or road bike for join health and muscle efficacy.
The sense of adventure, the sense of freedom and travelling and the feeling of speed might be very intriguing to some.
You should start with flat riding and slowly if wanted add hills.

So, one of the most important part of walking and cycling is the simplicity and the immediate availability. You can start them right now. I mean, can get your shoes, put on a podcast or audiobook and walk for 5hours. Get some salted water and off you go.

Weight lifting, callisthenics, crossfit. They are epic, they are great, however if you want to avoid injuries and possibly serious irreversible conditions, each movement need to be learnt profoundly. A simple squat is just so freckin' complex, that if you wanted to build up a series of movements from zero to hero, it would take serious amount of time. If you wanted to perform those against time, with speed, this learning curve is even steeper and longer.
What, I recommend is to learn slowly these and include slowly them into your walks or cycling. If you did not care, you should still do body weight strength as it is a different kind of metabolic booster than endurance.

Whatever you do and your goal is, my recommendation to weight loss and to regain health is to start out with cycling and walking outdoors. Take your time and the possibilities of podcasts and audiobooks to learn. Learn other sports, activities and about useful information, that you can use later on. Disconnect sometimes though.

I am a lean runner. 5 to 8% bodyfat. I have been this way for over a decade now. I am always in first 20, very often first 3 and sometimes winning races. I am on strava. I am competitive. However, I still do not dismiss the benefit of easy walking and cycling. I don't care if others pass me. Often friends ask me, what do you do out here ? What do you mean ? Well, they just don't understand the concept of chilling and being out for fun while not pushing myself. I am a mountain guy.
You don't need to care. The moment you decide that you will be out on the door, you become an athlete. It is not your clothes, your shape, your weight or others, but you, your mentality and discipline what decide.

We did not talk about swimming. Well, I find that it is not that good of a program as your heart rates are very low. It can be a great start for somebody extremely obese. Pools are often surrounded by saunas and ice-bathes. You can benefit from this too.

Walk or ride. The rest comes after !

Running / fasting for weight loss

Best sport activities for weight loss

Best sport activities for weight loss

Best sport activities for weight loss