First, since u and v are linearly independent vectors that span S, they are a basis for S.
To show that {u- v, u+ v} is a basis for S, you must show that they also are independent and span S.
So you must show that if a(u- v)+ b(u+ v)= 0 for numbers, a and b, then a= 0 and b= 0. You can do that by combing multiples of u and combining multiples of v and using the fact that u and v are independent.
To show that u- v and u+ v span S you must show that any vector in S can be written as a linear combination of u- v and u+ v. Of course, because u and v span S, any vector in S can be written as au+ bv.