Simplifying a lot: Winds generally blow from higher air pressure to lower air pressure. The winds blow the clouds along, and clouds cause rain.
So if there’s an area of low pressure, it’s probably going to be cloudy and rainy there, because the winds will gather all the clouds there. If there’s an area of high pressure, it’s probably going to be sunny there, because the winds will take all the clouds away.
Many years ago, just having a measurement of the air pressure where you are would let you do basic weather forecasting: If the pressure is high it’s mostly sunny. If the pressure is low it’s mostly cloudy and rainy. If the pressure is going up, the weather is getting sunnier. If the pressure is going down, the weather is getting rainier.
Nowadays, computer weather forecasting uses the air pressure as one of the many inputs into their very complicated models. It’s become much less relevant to people who are not weather experts, because we can just use the weather forecast without worrying about how that forecast was generated.
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