ELI5 explanation: They have a shitload of historical data. Weather tends to follow historical patterns, so what they do is to go “We have this situation, which historical situations matches this best?”. They then look at what those situations looked like a while later, does some kind of average of those historical results and calls that the prediction for, say, now +1 hour.
Then they repeat the process to move further into the future. Of course, the further you go, the more errors will add up and the less reliable the prediction will be.
This is of course vastly simplified, but it’s the basic principle.
I assume they’ll be using AI and pattern matching soon, if they aren’t already.
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