They arise in the representation theory of quivers: Gabriel's theorem says that a connected quiver has finite representation theory type if and only if it is of type ADE, and then the indecomposable representations correspond to the positive roots. This has been extended to symmetrizable Cartan types by Dlab and Ringel and to infinite root systems by Kac.

I think of this as ultimately deriving from the connection between root systems and crystalographic reflection groups (M Winter's answer): If $Q$ is a quiver, then the Grothedieck group of $Q$ representions is a lattice equipped with a bilinear form, and Bernstein, Gelfand and Ponomarev found "reflection functors" which act on this lattice by reflections.