What exactly is a model grid?
A model is a simplified representation of a system.
Rather than try to model everything down to the microscopic level, or as vectors, we often aggregate phenomena across geographic areas, to simplify the computation a lot, and to lose only a little accuracy and precision.
So if I want to model the UK, which roughly spans 8°W - 2°E, 50°N - 60°N, that's an area of 10° x 10°. I can split that up into 400 cells, each 0.5°x0.5°. So, starting in the south-west corner, I model the area 8°W-7.5°W x 50°N-50.5°N as a single homogenous area, with no variation within it. And the same for the other 399 cells (8°W-7.5°W x 50.5°N-51°N and so on all the way through to 1.5°E-2°E x 59.5°N-60°N). That would be a 0.5° grid.
Note that the gridcells do not overlap at all, and they completely cover the area of interest. (in mathematical terms, they are exclusive and exhaustive - every point of interest is in exactly one gridcell).
That can create vast amounts of data - particularly if I'm doing this for the whole world, at second-by-second resolution for ten years, for a few hundred variables of interest. And no one will ever need that amount of precision from my results. So I aggregate up to a more coarse grid: so that I take 25 grid squares that form a larger square, and aggregate each of my statistics of interest in them. So now my area 8°W-7.5°W x 50°N-50.5°N is just one of 25 cells in a square in the coarser grid which spans 8°W-5.5°W x 50°N-52.5°N, and I report statistics for 16 cells in a 2.5° grid, rather than 400 cells in a 0.5° grid.
Exactly how that aggregation is done, will vary for each statistic:

If it's a minimum overnight temperature, then I'll take the minimum across all 25 0.5° gridcells, to give the minimum overnight temperature for the 2.5° grid cell that consists of those 25.

If it's an average rainfall, I'll take an average.

If it's insolation per unit area, and I'm interested in insolation per unit land area, then I'll take a weighted average of insolation, weighting by the proportion of each 0.5° gridcell that contains land.


And then I'll have modelled on a 0.5° grid, and archived on a 2.5° grid
Here, for illustration, is Britain in a 100km grid:
 source
If I wanted to aggregate that to a 200km grid, then the four squares in the bottom-left corner, SQ, SR, SV, SW, would be combined into a single grid cell. And SS, ST, SX, SY would be combined into another grid cell. And so on.