Ok, maybe a little multimedia presentation will sort this out.
Force due to gravity is an inversely affected by the square of the distance between the center of mass.
Now you could view the earth as a number of really really small but equal masses; think really really small pieces of rock and each is a separate object that just happens to be placed really really close to other really really small rocks.
But, if you assume the Earth is a sphere of constant mass (which it isn't), you can combine all those smaller masses into one big mass and the math will work out the same. Plug in some simple values and you will see that this is true.
However, things get a bit tricky once you place one object within the other, because now you have to start looking at it as a bunch of smaller masses again. This is because the entire mass of the one object is no longer "pulling: the second object in the same direction. Simply illustrated, it would be like placing one object equidistant from two identical masses.
Now, if we extrapolate this to the idea of an object at the center of he Earth, and treat the Earth no as one large mass, but as an infinite number of smaller, but equal, masses you could visualize it like this. (not to scale)
Each one of those masses wants to pull the poor balloon towards it, but since they are all theoretically the same mass, and the balloon is hypothetically in the exact center of the smaller masses, the balloon will stay where it is.
I have oversimplified things and ignored a number of other factors...
The Earth is not a true sphere
The Earth's density and mass fluctuate from point to point.
The extreme heat and pressure at he center of the Earth would render this moot.
The object placed at the center of the Earth would also have to be a perfect sphere for this to work.