We consider what happens to the capital stock when the savings rate increases:

The economy begins in a steady state with savings rate $s$ of 0.18 and a capital stock $k$ of 1. When the savings rate increases, the $s f(k)$ curve shifts upward. At the initial savings rate, the amount of investment $s f(k)$ just offsets the amount of depreciation $\delta k$. Immediately after the savings rate increase, investment is higher but the capital stock and investment are unchanged. Therefore, investment exceeds depreciation:

The capital stock gradually rises until it reaches a new steady state, with a higher capital stock and higher level of output than the previous steady state.

The saving rate $s$ is therefore a key determinant of the steady-state capital stock. If the saving rate is high, so will be the capital stock $k$ and output in the steady state. If the saving rate is low, in the steady state the economy will have a small capital stock and a low level of output.

What about the relationship between saving and economic growth? As the savings rate increases, so does output growth but only temporarily. As the economy converges towards the new steady state, output growth falls to zero. Hence, if the saving rate is high, the capital stock will also be high, but it will not maintain a high rate of output growth forever.