Because it is made of rubber.
Natural rubber is made from the sap of the rubber tree. It is called latex, and it is a white sap similar to that of milkweed plants and dandelions. The word gum refers to the sap of plants, and gums are used for sealing envelopes and gluing things together.
Bubble gum and chewing gums these days are made from the sap of the manilkara chicle tree. The sap itself is just called chicle. It was once used as a substitute for natural rubber, but is now mainly used for chewing gums. The name for one brand of chewing gum, Chiclets, comes from the word for the raw sap.
A cheaper form of synthetic rubber is now used in many chewing gums.
Rubber and chewing gum are both examples of polymers, long chains of atoms strung together. The strings are tangled together, and occasionally cross-linked so that one strand is connected to another, often in several places. This is what makes the rubber bounce back, or pull together after being stretched. It is what gives something a rubbery, stretchy, feel.