In the book "Essential mathematical methods for physicists" from Weber and Arfken, they define the integral form of the gradient,divergence and curl, althougth they give sections before an explanation of the upper parts of the divergence and curl definition, I don´t understand where does the upper part of the gradient definition come from and also I don´t understand why the integral of $d\tau$ and its limit appears in all the three definitions.
Imagine that you are integrating $\phi$ over a box of side $a$ with center at zero. Then, the integral would be aproximately given by $$\int \phi d\sigma\approx A(\phi(a,0,0)-\phi(-a,0,0),\phi(0,a,0)-\phi(0,-a,0), \phi(0,0,a)-\phi(0,0,-a))$$ where we are assuming that the box is sufficiently small s.t $\phi$ doesn't change over any of its faces, and $A$ is the area of them. You can see that as $A$ goes to zero so does the integral. But, if you divide it by the volume of the box, you end up with $$\frac{1}{a}(\phi(a,0,0)-\phi(-a,0,0),\phi(0,a,0)-\phi(0,-a,0), \phi(0,0,a)-\phi(0,0,-a))$$ which in the limit $a\rightarrow 0$ is of course the gradient.
This hopefully makes clear the role of the volume in the denominator. We divide by it because otherwise those "infinitesimal changes" we want to study just amount to zero in the limit. You can think of it like the various densites. For example, if you try to study the mass of a liquid and want to talk about the mass at a single point, you invariaby end up with zero; but if you divide the amount of a mass of a given region by its volume and make the volume go to zero you get a useful non zero quantity defined at a single point that stills gives you a notion of "mass at a point".These definitions are exactly like that.