Edit. Even better:
https://en.wikipedia.org/wiki/Numerical_weather_predictionVery broadly, I'm not an expert in this.
It's simulation.
The models for wheather split the atmosphere in 3D blocks, on to which observed data parameters (temp, pressure, wind direction & speed, humidity, more perhaps?) is "projected".
Then calculations are run to forecast the behaviour of each and every block and its neighbors, on the n next time steps.
The smaller the blocks, the more precise you can forecast.
The more the blocks, the larger the area you can cover.
Each and every parameter can impact others.
Each and every cell impacts its neighbours.
All of this combines into quite a huge computational challenge.