Is it right that in order to lose body fat you must exercise?
To lose body fat you must create a calorie deficit. You can create that deficit by
1. Exercising, which will improve your overall health and fitness.
2. Eating fewer calories
Even someone who is injured can lose body fat, despite a lack of exercise. The complaint "I gained weight whilst I was injured because I wasn't able to exercise" could more correctly be stated as "I gained weight while I was injured because I mindlessly overate for comfort and fun".
Adding exercise to your life does not necessarily mean losing body fat at a significant rate, you may exercise 3 times a week for an hour, you may burn 600 kcals each hour adding up to 1800 a week but this may only be 900 kcals of body fat, it will take you 4 weeks to burn enough calories to lose 1 pound. this is also relying on you not falling into a habit of eating more because of the exercise. Or you will have to quadruple your effort, 12 hours of exercise a week!
There is also a possibility that the hours not exercising are sedentary, you are recovering, your legs are sore, you've worked hard so deserve a break. This all points towards less calories being used during your time not exercising.
You are someone who is active, not someone who exercises. Use every opportunity to burn some extra calories, walk wherever possible, do the gardening, take the stairs, be as active as you can all of the time.
Creating a deficit of 500 kcals in your diet every day may lead to a weight loss of 1 pound a week.
Do both, exercise to be fit and healthy and limit calorie consumption to get the most that you can out of life, following such a regime gives you so much more opportunity to lose body fat.