Why is the potential energy equal to the negative integral of a force?

When you do conservative work on an object, the work you do is equal to the negative change in potential energy $W_c = - \Delta U$. As an example, if you lift an object against Earth's gravity, the work will be $-mgh$. Gravity is doing work on the object by pulling it towards the Earth, but since you are pushing it in the other direction, the work you do on the box (and therefore the force) is negative. The field does negative work when you increase a particle's potential energy.

Mathematically, it is just that $F=\frac{dW}{dx}$, which means that if the work is conservative, then $F=\frac{-dU}{dx}$, since $W_c = - \Delta U$. Then $-dU = Fdx$, so $U = - \int F dx$.

We can also say that work is negative when the force and displacement are in opposite directions, since $W = \vec F \cdot d\vec x = Fdxcos\phi$. When $\phi=\pi$, then $\cos\phi = -1$. An example of this conceptually is friction. An object sliding down a plane has kinetic friction acting on it. The friction is in the direction (up the ramp) opposite to the object's motion/displacement. So we say that the friction force is doing negative work.

It is all about finding or constructing conserved quantities.

When an object is under forces, in general the KE of the object is no longer a constant. But can we add something to it so that we have a conserved quantity again?

People derived that by Work-KE theorem $$\Delta KE = \int_{t_i}^{t_f} \textbf{F}_{net} \cdot \textbf{v} dt$$ where $$\textbf{F}_{net}=\textbf{F}_1+\textbf{F}_2+\cdots$$ is the net force acting on the object.

Then we found that for some force $$\int_{t_i}^{t_f} \textbf{F}_{k} \cdot \textbf{v} dt = \int_{\textbf{r}_i}^{\textbf{r}_f} \textbf{F}_{k} \cdot d\textbf{r}$$ which is path independent and called conservative forces. Forces don't satisfy this property are called non-conservative forces.

So we want to "move" these terms to the LHS and we have $$\Delta KE - \int_{\textbf{r}_i}^{\textbf{r}_f} \sum_{conservative} \textbf{F}_{k} \cdot d\textbf{r} = \int_{t_i}^{t_f} \sum_{nonconservative} \textbf{F}_{k} \cdot \textbf{v} dt$$

So we have the minus side because we "moved" them to the other side of the equation.

Now if we define $$PE_k(\textbf{r}_f)-PE_k(\textbf{r}_i)=-\int_{\textbf{r}_i}^{\textbf{r}_f} \textbf{F}^{conservative}_k\cdot d\textbf{r}$$ then we have $$\Delta KE + PE_1(\textbf{r}_f)-PE_1(\textbf{r}_i) + PE_2(\textbf{r}_f)-PE_2(\textbf{r}_i)+\cdots = \text{Work done by nonconservative forces}$$

If there are no nonconservative forces, or when the nonconservative forces do no work, then we have the conservation of energy, where total energy is defined as the sum of KE and PE.

Note that if you are fine to accept the total energy being KE - PE, then it is completely fine to define PE without the minus sign.

As for your last question, you can imagine that you apply a force which is just "slightly" larger than the conservative force. Then the object will move very slowly. When it is close to the final position, reduce your force so that it is just "slightly" less than the conservative force so that the object will slow down.

The potential energy (PE) is the kinetic energy (KE) that you could get out of an action if it were to go ahead and happen. The energy you put into a ball to move it up 10 feet is the same amount of energy you could get back out if you let it return to its starting point (under similar conditions).

***Note the words put in and get out. One of these actions will be considered negative, and the other positive. It doesn't matter which as long as you stay consistent with your signs.

From this example, the work done on the ball to move it up is the amount of energy it took to get it up there. And by definition, work equals the integral of force over distance. -> If we say that the energy we put in to the ball is positive work, then energy that we can get out of the ball will be negative work.

Or, instead of saying negative work, we could say its the work done by an opposite force to the original. Which leads to the integral of the negative force.

I hope this helps!

edit: slight wording adjustment