$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$
These components are the rate of increase along the $x$, $y$ and $z$ directions respectively and according to gradient operation maximum increase is along the axis.
Why does it have to be along the axis? Why does maximum increase have to be along the $x$, $y$ and $z$ direction?
Let's say we calculate the gradients at point $(x_0,y_0,z_0)$. We are interested in the rate of increase of $f$ in the direction of some vector $(x_1, y_1, z_1)$, and let's say that the vector has norm $1$.
Let's have a single variable function
$$g(t)=f(x_0 + tx_1, y_0 + ty_1, z_0 + tz_1)$$
and let's find $g'(t)$.
Using the chain rule, you can verify that
$$\frac{dg}{dt}(t) = \nabla f(x_0 + tx_1, y_0 + ty_1, z_0 + tz_1) \cdot (x_1, y_1, z_1)$$
So the rate of change at $t=0$ is
$$\nabla f(x_0 , y_0 , z_0 ) \cdot (x_1, y_1, z_1)$$ This inner product is maximum (if the norm of the second vector is $1$) when the two vectors are paralel, so the conclusion is that the rate of change is biggest in the direction of $\nabla f(x_0,y_0,z_0)$