Let M be a cyclic R module, with a generator g. Let H be the left ideal in R that maps g to 0. Thus M is isomorphic to the cosets of H in R. In other words, M is a quotient module of R, and we know that the homomorphic image of a semisimple module is semisimple. Therefore M is semisimple.
Remember that a module is semisimple iff it is spanned by simple modules.
Now let M be any R module. Each element of M acts as a generator, spanning a semisimple module. Yet every semisimple module is spanned by simple modules. Therefore all of M is spanned by simple modules, and M is a semisimple module.