my bad, i completely forgot about this. i hope this still helps.
i'm pretty sure S is the subspace itself here.
so the first thing it's telling you is that S is a set. that means there are things in S (nonempty), and those things are vectors (in Rm).
then, the definition gives you two axioms ("subspace axioms"). what that means is that a subset fulfilling these conditions is a special kind of subset (a subspace).
so in order to show that S (which you already know is a subset by definition) is a subspace, you just show that for all the vectors in S (which are also in Rm, which S is a subset of), (1) and (2) are true.
so all a subspace is is just a type of vector space, that's "closed" under the two axioms and contains some vectors in Rm. you kinda just have to memorize that.
hopefully this helps. any more questions, i'll try to answer or clarify.
