Why doesn't the supply curve shift to the left in the following situation? Why do producers respond to the surplus by cutting their prices?

Why doesn't the supply curve shift to the left in the following situation?

Because in order for a demand curve to shift demand has to change at any price. For example, let us suppose that the demand function is given by:

$$D_1 = 100 -P$$

If a firm changes price it chargers it does change quantity demanded $D_1$ but does not shift the whole demand function $D = 100 -P$. If quantity demanded changes just because price varies, we call this movement along demand curve.

However, suppose now that there is some change in preferences of people so that demand function $D_1$ shifts and we will have new demand function $D_2$:

$$D_2 = 50 -P$$

Can you see the difference? Now at any price demand will be smaller compared to $D_1$. If price is $5$ the demand in $D_1$ will be $95$ but in $D_2$ only $45$, if price is $10$ then $D_1 = 90$ and $D_2 = 40$ and so on. This is what we would call shift in demand (in fact if you would plot $D_2$ an $D_1$ you would see that $D_2$ is just $D_1$ shifted to the left.

Why do producers respond to the surplus by cutting their prices?

Because, within the bounds of a simple supply-demand model, if there is a surplus of goods producers have two options:

A) throw the goods away and get nothing.

B) sell them at lower price.

selling them at lower price if you can is more rational then just throwing the goods away (at least it would be in a simple models, there might be some complex strategic interactions where it might be other way around, but you won't encounter such more nuanced and complex models in econ 101 textbook).