@abealti I think it would help understand better if you gave us a bit of background - what type of diabetes do you have, how long have you had diabetes, how many carbs do you eat, what are your typical BG levels?
I assume you have type 1 diabetes based on your insulin regime (you mention that you stopped taking insulin with meals but continue with your long acting insulin), so if you have not had diabetes for long, you could be in the honeymoon phase whilst your body is still creating some insulin. For many people with type 1, the honeymoon phases can be unpredictable - we may need no insulin one week and suddenly our pancreas goes on strike the next week.
Whilst you are eating carbs, if you are eating a small amount of low GI carbs, your background insulin may be able to cover your insulin needs whilst exercising.
When you exercise, your body does two diabetes related things - your liver will release extra glucose to give you energy and your body becomes more efficient at using insulin. Depending on the type of exercise, your fitness and the length of exercise and your BG before you start exercising one of these may dominate so your BG may go up or down.
For example, when I climb (short, strength, resistance training) my BG goes up whereas when I go to a spin class my BG goes down. If my BG starts too high, I struggle to exercise (I have less energy, my legs feel like lead) and the stress increases my BG.
I cannot comment on fat burning as my body weight has always been stable. However, when I do more resistance training, I certainly build more muscle. This has nothing to do with diabetes or insulin intake as this happened before I had diabetes.
If you are maintaining good BG, and are able to maintain the level of exercise needed to do this, I think (but I am not a doctor) this is ok. However, as you get fitter, the impact on your BG may reduce so you will either have to exercise more or take some insulin.
Whatever you do, do not stop taking your long acting insulin and do not stop monitoring your BG.