Let's imagine discretizing spacetime on a lattice. For an ordinary scalar field, you can independently the field value at each point on the lattice. Thus there is one degree of freedom at each spacetime point. We say that the field is a local degree of freedom. If you like, the number of local degrees of freedom is the number of field values you specify on the lattice, divided by the number of lattice points.
To discuss energy, let's take a simple example where we have a field sitting at the minimum of the potential. Then the energy associated with a single lattice point is determined only by the potential energy $V(\phi_{min})$. Here we see that the energy density is fixed over the whole spacetime, in the sense that the energy associated with each lattice point is the same. If there are $N$ points on the lattice, then the total energy of the whole lattice is $NV(\phi_{min})$. The energy scales like the number of spacetime points, which is the volume.
Based on the holographic principle, the expectation is that quantum gravity won't have local degrees of freedom. For example, we expect the number of degrees of freedom in a region to scale like the area of that region, not its volume.
