Investment dollars need to be borrowed from somewhere. That comes from saving. So at any point of time, one can argue S=I. If at that instant of time, government suddenly decides to borrow more, the private sector will need to do with less.
However, we are concerned with the temporal movement of savings. That is how do savings S1 at time T1 change to savings S2 at time T2.
Suppose the govt decides to do nothing. There is an implosion in the economy. The govt did not borrow at time T1, yet because of the economic implosion, savings S2 at time T2 are 0 (for argument's sake), so the private sector can't borrow at time T2.
On the other hand, if the govt does something and the economy doesn't implode, savings S2 will be higher than in the scenario above. While there will be government borrowing, because of higher savings, the private sector too will be able to borrow something.
So basically, govt intervention should be such so that savings S2 at time T2 are higher than without government intervention.