What I Know: $$\vec F = -k \vec x$$

where the negative sign indicates the Force acts in the opposite direction to the displacement.

If we were to take the integral so...

$$\int_{x_i}^{x_f} Fdx = -\Delta U$$

What would the negative sign in this instance represent? From my understanding, we cannot produce negative energy...or can we? I have attached the image below for the context of my confusion. Thank you.

See first of all Potential Energy arises when work is done against a conservative force.This means that:

Change in Potential Energy = - Work done against conservative force $$ΔU = - W$$ Thats where the negative sign comes from

This means that work done against conservative force (or its negative value ) is equal to change in Potential Energy.

From my understanding, we cannot produce negative energy...or can we?

What you are missing here is that we are not producing 'negative energy' but negative change in energy.

The change in Energy may be positive or negative or zero.

Notice how I have stressed on change.

P.S. If you still have any doubt,comment below.

The -kx is the force exerted by the spring as it is stretched (or compressed) away from the equilibrium position. Your integral is the work done by the spring (which is negative when being stretched). To get the increase in energy being stored in the spring, you need the work done by an external force.