Because it describes spinors (i.e. behavior of fermions). String Theory speaks about both bosons and fermions... when passing to a QFT, fermionic wavefunctions become spinor fields, and their dynamics require the Dirac operator. And this is where all the mathematical importance stems from: QFT and String Theory (topological field theories) is a pure math theory. In particular, I disagree completely with Noldorin's seemingly naive comment... Clifford algebras and spinors on manifolds is a huge field.
Sorry if this too brief of an explanation (for immediate purposes).
This post imported from StackExchange Physics at 2014-04-01 12:38 (UCT), posted by SE-user Chris Gerig