Infinite is a big word :) . As explained above, the extra spacial dimensions would need to be compact, which implies them being finite in size. Perhaps the simplest example of compact manifold is the circle: you can imagine one extra dimension to be closed on itself. Going around the circular dimension one circumference length, you will find yourself where you started. The size of the dimension then refers to the characteristic size of the circle, i.e. the circumference.
