I'll give you some advice on how to do proofs.
When you want to prove something satisfies a definition, then you need to be familiar with that definition. In this case, you need to know the definition of a subring. From what I vaguely recall, S is a subring if it is also a ring and it has the same multiplicative identity as R (I'm not totally sure on this though).
So how do you start the proof? Well, start with the assumptions in the theorem. In the first direction you have "S is a subring of R" implies $\Rightarrow$ "S is nonempty, S is closed, etc."
This part is easy. HINT: Look at the definition of a subring.
Next, since this is an "if and only if" ($\Leftrightarrow$) statement then you have to also show that it works from the other direction:
S is nonempty, S is closed, S contains negatives" implies $\Rightarrow$ "S is a subring of R"
So work with your assumptions (S is nonempty, S is closed, etc) and figure out how these assumptions alone are enough to show that S satisfies all the other requirements of being a subring (which you will find in the definition of subring).
I would provide more detail but, like I said, I don't remember the definitions of rings or subrings off the top of my head.