What exactly does it mean for a conservative force $\vec{F}$ to not depend on the time or velocity, only position? How does this imply that the line integral is independent of the path taken?

The condition $\vec{F}= \vec{F}(\vec{r})$ says that the force direction and magnitude only depend on the position $\vec{r}$ at which it is applied. The force does not depend on time (i.e., it does not vary in time-- the force will be the same now as it was five minutes ago), or on velocity (the force is the same no matter how fast the particle is moving), or any other variable. For example, gravity near the surface of the Earth satisfies this condition. The force of gravity is different for different distances from massive objects-- it depends on $\vec{r}$. However the force of gravity does not depend on time, and it does not depend on how fast a particle is moving.

Note that the condition that $\vec{F}$ only depends on position does not imply that the line integral is independent of the path taken-- these two are independent conditions, and both must be satisfied in order for a force to be conservative.