The dimension of the worldsheet of a fundamental string is two. The maximal spacetime dimension in which two worldsheets generically cross - and allow the wrapped strings to annihilate (and unwind) which allows the space to expand - is "2+2=4", which is why we are supposed to live in four large dimensions. A larger number of dimensions can't expand because the strings would not have enough chance to annihilate (there is too much space and the strings have too a small dimension), and the remaining wrapped strings would prevent the small dimensions from expanding to astronomical sizes.
You might object that this argument also allows a lower-dimensional spacetime to develop, but 3+1 dimensions are preferred. If nothing prevents 3 spatial dimensions from expanding, it will occur.
There is a huge number of small problems and subtleties about this proposal, but it is undoubtedly attractive to imagine that a cosmological mechanism explains things such as the dimensionality of the Universe. Recently, in the context of the second superstring revolution, the Brandenberger-Vafa framework was upgraded to the "brane gas cosmology" which includes not only strings but also higher-dimensional branes.
Lisa's approach is the opposite one: she wants to consider the mutual annihilation of branes in a higher-dimensional Universe. Consider some simplified type IIB string theory. The branes with low dimensions won't annihilate too much, but the p-brane energy density will simply decay as
Many formulae looked more concrete than last time when I was collaborating on this line of reasoning, and as Lisa says, this sort of Brandenberger-Vafa reasoning may be useful to identify a cosmological selection mechanism that will find some preferred vacua (in this case: braneworlds) within the landscape of possible vacua.