The core idea behind this type of approach ("parametric encoding") is that you learn a scene as some spatial data + a (small) neural network. For example, a 128^3 grid of data values and a 10k parameter model. In the forward pass you feed whatever data is at the voxel(s) in question to the network, and the backward pass updates both the network and the same voxel(s).
The innovation in this paper is in how the spatial data is represented. Prior work includes dense grids, multi-resolution grids and octrees to name some - but all of them are either GPU-unfriendly or waste parameters on empty space. They figured that they can just hash the coordinates and use them directly as an index into a data array (edit: A multi-resolution stack of data arrays - sorry for not getting this right initially), with hash collisions left to the network to figure out (it's gonna figure out whether there's a collision on fine layer through info from the coarser ones, I guess).
(Relatively) few parameters + GPU-friendly data structure = fast training. Tempted to try and implement this myself...