The reason a balloon filled with air pops when you put it over a flame is because the rubber of the balloon gets very hot and weak and then breaks. When you fill a balloon with water instead of air, the water absorbs most of the heat, so the rubber doesn't get very hot. Water has a very high heat capacity, much higher than air, so it will keep the balloon from bursting.

When you put an object over a flame, it starts getting hot-- gaining heat that is released from the flame.In a normal balloon filled with air, the rubber of the balloon quickly heats, fraying and coming apart-- so the balloon gets a hole and pops. But if the balloon is filled with water, the heat that the balloon rubber is gaining can dissipate into the water inside, raising the temperature of the water and leaving the rubber surface cool and undamaged. This is because water transfers heat much more efficiently than air, and has a higher heat capacity (how much heat is needed to raise its temperature).

Conduction: it takes a lot more energy to heat up a volume of water than a volume of air, and so the heat from the candle just goes into the water and doesn't warm it up much. If the balloon is filled with air, however, the heat goes into the rubber, which causes it to fail and burst the balloon.

The skin (plastic) of the balloon is very thin, so it gets cooled by the water inside. If you held it over the fire long enough, eventually the water would reach a temperature that was too high for the plastic to survive, and then it would pop.

Water absorbs most of the heat from the flame so that the surrounding rubber cannot increase its temperature enough to melt (let alone burn).