Don't worry, this doesn't mean you have to become a muscle-bound bodybuilder. However, by exercising, you convert fat into muscle. If you have more muscle, you burn more calories, and even at rest. The body's basal metabolic rate, the amount of calories the body needs to maintain all its functions, increases with every single gram of muscle mass.
The human body begins to lose muscle mass as early as the age of 25. Statistically, the body then loses three kilos of muscle in every decade of life. This also reduces the basal energy metabolic rate by about 100 calories per day. The result: you put on weight. By the way, you build muscle not only through weight training, but also through exercises with the swinging bar or rowing machine and on the cross trainer. So pick exercises that you like! Do the exercises with light weights and do more repetitions for them - this way you'll build muscle without gaining volume. Two to three workouts per week are ideal; with additional endurance sessions, you can increase your overall metabolism even more.
But remember: muscles are heavier than fat. Therefore, you may even gain some weight during the weight loss phase, but you will still lose girth and look slimmer. That should count for you! So your training goal should be: More muscle, less fat.