Why is a conservative force defined as the negative gradient of a potential?

We introduce a minus sign to equate the mathematical concept of a potential with the physical concept of potential energy.

Take the gravitational field, for example, which we approximate as being constant near the surface of Earth. The force field can then be described by $\vec{F}(x,y,z)=-mg\hat{e_z}$, taking the up/down direction to be the $z$ direction. The mathematical potential $V$ would be $V(x,y,z) = -mgz+\text{Constant}$ and would satisfy $\nabla V=\vec{F}$. This would correspond with decreasing in height increasing in potential energy which would make us have to redefine mechanical energy as $T-V$ in order to maintain conservation.

Instead of redefining mechanical energy, we introduce the minus sign $\vec{F} = -\nabla V$ which equates the physical notion of potential energy with the mathematical notion of the scalar potential.

If the resultant force acting in a body is give by minus the gradient of potencial you can show that $\frac{dE}{dt} = 0$. Where E is the total energy of particle. So total energy, kinect + potencial is conserved.

In 1-dimensional case:

$\frac{dE}{dt}=\frac{d(\frac{1}{2}mv^2+V(x))}{dt}=mv\dot{v}+\frac{dV}{dx}\frac{dx}{dt}$

$=v(ma + \frac{dV}{dx})$