EDIT: See /u/jaa101’s posts below. While this explanation is partially correct, it’s wrong in at least a couple critical ways.
The mass isn’t the relevant factor, the *density* (mass per volume) is. Something less dense than air floats, something more dense than air sinks, something much more dense than air (like most solids) falls.
When the balloon is inflated with a light gas like helium, the balloon (not just the plastic skin, but the skin *and* the gas inside*) is not very dense. Most of its mass is the skin of the balloon, which effectively gets spread out over a much larger volume by being inflated by something with very little mass. So the density is lower. When it deflates, the mass stays close to the same (there’s some lost from having less gas inside it, but the gas wasn’t contributing much) but the volume shrinks dramatically, which makes the density go up.
The same principle allows boats (metal hull + lots of air inside = lots of volume to spread the weight of the hull out) to float on water, but lets them sink easily if their hull gets punctured by something (because then it’s just metal hull spread out over its *own* volume, which is heavier than water). The same laws apply for both.
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