Archimede's principle is a principle of buoyancy.
He noted that all objects placed in water apparently lose an amount of weight equal to the weight of the water displaced by the object.
The balloon contains, in its envelope, a gas lighter than air, either hydrogen or helium, or equally some hot air.
One cubic meter of helium can lift 1000 grams, one cubic meter of hot air at 100°C can lift approximately 200 grams.
Therefore for the same volume, a helium balloon can lift five times the weight of a hot air balloon, but the principle is the same.
More the balloon's volume is increased, more the balloon is able to lift.
Air bubbles in the water has an equal traction of weight as the displaced air that allows the balloon to rise.
The same principle allows a boat to float.