Since, I was urged to present an answer for didactic reasons to which I totally agree, I will provide the full set of corrections to avoid any ambiguity.
% Model parameters
alpha = 1/3; %
s = 0.2; % Investment rate
delta = 0.2; % Depreciation rate
n = 0.02; % Growth of labor force
g = 0.01; % technological progress
eps = 1;
k(1) = ...
I don't know how this can be answered without any recourse to at least basic mathematics, since the question itself refers to movement along some curves which is mathematical concept. Hence I will go over some math and then provide intuition at the end.
Explanation Including Math
The investment curve in Solow model is defined as $sf(k)$ where $f(k) = Y$ and ...
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 ...
Savings in macro are equal to investment $(S=I)$. Investment is how you get new capital stock. When people save more they also by definition invest more and when they invest more there is more capital.
They used a product rule, quotient rule and chain rule, I won't be discussing the rules themselves as such topics belong to Mathematics.SE, but the reason why you might not realize this outright is the particular notation in a growth theory, which would be on topic here so I will focus on that.
In Solow growth model with labor augmenting technological ...