It's a bit unclear from the question, but I'm assuming you're asking why the tubular "bridge" remains straight, rather than bending under the force of gravity (as an un-inflated balloon would do).
The basic reason is because the pressure in the balloon wants to keep the walls of the tube as far apart as possible (even to the point of stretching the rubber). In order for the tube to bend, the walls would have to come together and fold over as a kink in the tube. As long as the force from the pressure differential which holds the walls apart is greater than the gravitational force exerted by the (quite light) end of the tail, it will not collapse.
Alternatively, the walls could stay inflated and the rubber would simply stretch to make the tube bend over. In this case, the gravitational force would have to overcome the elastic force of the rubber.
In either case, if you were to add some weight to the end of the tail (e.g. by taping a coin to it), you would certainly observe one of these two failure modes, as the weight overcomes the pressure and/or elastic resistance.