Why does a rise in the savings rate result in an increase in the capital stock intuitively? I understand mathematically how it works. But what is the actual process intuitively.
Thanks
The story in the other answer is not fundamentally wrong but incomplete and bit inaccurate.
Saving does actually affect capital stock through investment in Solow model (assuming based on the Solow tag that is the model you want intuition for), but there is more complexity to it.
What is saving
Investment is actually equal not just to private saving but both private and public saving since saving is actually defined as (following Blanchard et al Macroeconomics: a European Perspective pp 51):
$$S \equiv Y - T-C$$
Combining this definition with output identity for closed economy $Y\equiv C+I+G$ gives us:
$$S = I +T -G$$
However, it is often convenient in Solow model to just ignore government by assuming $G=T$, in which case $S+I$.
Why does saving rate increase capital stock:
Next the way how savings rate affects capital accumulation is depicted in the picture I took from Romer Advanced Macroeconomics 4th ed pp 19.

As you can see the intuition really works this way. Here $s$ is the savings rate and $f(k)=Y$. Higher $s$ means that savings increases, for example if $Y=100$ with $s=0.5$, $S=50$ and with $s=0.7$, $S=70$. Then if we assume away government indeed $S=I$ so $I=70$.
This is why the evolution of capital accumulation (per unit of effective worker) is given by:
$$\dot{k}(t) = sf (k(t)) − (n + g +\delta)k(t)$$
where actually $sf(k(t)) = sY = S =I$, so it is through increased investment. The $n$ and $g$ would be population and technological growth respectively and $\delta$ the rate of depreciation (although these are not important for intuition about raising level of capital).
Does this push capital beyond steady state?
Now this is surprisingly common misconception. Increase in savings rate can increase the level of capital stock - it just cannot increase the growth rate at which new capital per effective worker accumulates.
According to Romer Advanced Macroeconomics [emphasis mine]:

, a change in the saving rate has a level effect but not a growth
effect: it changes the economy’s balanced growth path, and thus the level of
output per worker at any point in time, but it does not affect the growth
rate of output per worker on the balanced growth path.

However, this is quite common misconception because of the confusing terminology. Increase in level in macroeconomic. Consider for example the visualization from Romer Advanced Macroeconomics pp 20:

The upper graph shows you the growth rate of capital per effective worker. Indeed here savings rate has only one-of impact that quickly dies out (in fact this is just level increase in disguise as growth).
However, on the bottom panel you can clearly see that the level of capital per effective worker is now higher thanks to higher savings rate.