If the balloon was completely isolated from the outside atmosphere and contained a certain gas (in this case helium), then the helium will exert a pressure outwards, attempting to expand the balloon. The atmosphere outside will also exert a pressure inwards, attempting to keep the balloon in a small shape. These two forces compete, and the balloon will always form a shape that exactly balances these two pressures, i.e. the net pressure acting upon the balloon surface is 0.
The only problem is that atmospheric pressure (the pressure outside the balloon), decreases with altitude, as shown in this graph supplied from Wikipedia:
Because the atmospheric pressure DECREASES as altitude increases, that means the pressure pushing the balloon inwards decreases, in other words as the balloon rises, the balloon will begin to expand outwards. Eventually, the balloon will expand until the expansion tears a hole in the balloon, and the bomb falls back to earth (not good). Therefore, there must be some regulatory system attached to the balloon, who's primary purpose is to regulate the pressure difference between the helium inside the balloon and the atmosphere outside (the exact mechanism can vary), so that the balloon won't explode, and can float away (though not indefinitely).
Most high-altitude balloons are only partially inflated at launch-off, so that they can have a lot of room to expand before popping:
Out of curiosity, what anime?