How is basal metabolism related to calories?

A calorie is just a measure of energy (heat). What we usually call calories are actually kilo-calories or large calories; one of them is the amount of energy required to raise 1 kilogram of water 1 degree C.

A metabolism is all the chemical processes of a living organism that produce energy, such as the processes of digesting and assimilating foods.

Therefore, the higher the metabolic rate, the more calories are being used by the organism.

To apply this to the weight of your body, all things being equal, the higher your metabolic rate for a period of time, the more calories you will burn. So, to lose weight, raise your metabolic rate; to gain weight, lower your metabolic rate.

You can lose weight by ingesting fewer calories, burning more calories by being more active, or both. Actually, it's not quite that simple, but that's the general idea. (The timing of ingesting calories turns out to be important, and different sources of calories are easier or harder for your body to utilize.)