As far as I'm concerned, the multiple dimensions used in string theory have a purely mathematical origin. The problem does not arise when looking at interactions, but when trying to make "classical" string theory compatible with quantum mechanics. Adding interactions only makes things more complicated. When you do the math, you get a (more-or-less) inconsistent theory, unless the number of dimensions of spacetime is exactly right. AFAIK there is no physical argument that explains or even makes plausible why there should be so many dimensions.