It is said that Archimedes discovered this principle while taking a bath and feeling easier that he normally ways: a body submerged into a fluid (a liquid or a gas) is pushed upside by a force equal to the weight of the mass of the fluid it displaces. He later called this force buoyant force.
Thus, a balloon that is inflated and is having a volume V displaces a mass of air having a weight (buoyant force)
`F_b =G_(air) =rho_(air)*V`
Because the balloon is inflated with a gas having a lower density than air (let say Hydrogen) the true weight of the balloon is
`G_("balloon") = rho_(H)*V`
Since `F_b >G¨_("balloon")` , `F_b` pushes up and `G_("balloon")` pushes down ` `there will be a resultant force that pushes up the balloon.
`F = F_b -G_("balloon")` is directed upwards.
The balloon is rising because the resultant force between the buoyant force and the true weight of the balloon is upwards.