Some people will save more and some people will save less. When interest rates increase, consuming now becomes more expensive relative to consuming in the future, because you have give up more future consumption for each unit of current consumption. This is the substitution effect, and it holds true for both borrowers and savers. On the other hand, the increased interest rate means savers have more total income, and so are better off. Therefore, the income effect leads them to consume more in both periods.
Borrowers pay more per dollar borrowed, and thus have less total income, so they are worse off.
The income effect leads them to consume less in both periods. So, when interests rates go up, savers are better off, but borrowers are worse off.