When the bottle is heated, the air molecules inside start moving faster. These molecules now collide into the balloon with more energy resulting in increased pressure. The increased pressure causes the balloon to expand.
The expansion is due to the elasticity of rubber. As we keep blowing, the air pressure builds up inside the balloon, and the rubber expands to its maximum limits. If we blow again, the balloon won't be able to expand but the air that we blows in need space still, and the balloon bursts.
There is no direct relation between mass and pressure of gases. An equation that involves the relationship between mass,pressure and volume of gases can be seen in Boyles Law:-
"The absolute pressure exerted by a given mass of an ideal gas is inverselyproportional to the volume it occupies if the temperature and amount of gas remain unchanged within a closed system. Pressure multiplied by volume equals some constant ."