There's an argument in string theory called the brandenberger-vafa mechanism that I think is pretty clever:
Imagine that a bunch of particles and antiparticles travel along in one dimension. Because they can't spread out, they'll eventually collide and annihilate. However in 2d that's not the case: they're more likely to just miss each other all the time.
If you increase the dimension of the "particles" to strings, you can make a similar argument. In 2 (compactified) dimensions, strings will find each other but not in three.
Then if you start with some compactified space of 10d with a bunch of strings wrapped around various 2d subspaces, they will naturally constrain that space so that it can't expand. However, as strings move around randomly they'll tend to collide and unravel, allowing various 2d subspaces to cease being constrained and therefore expand.
If you analyze this carefully you can show that in fact three of the spatial dimensions will tend to expand while the six remaining ones will remain compactified.