Weather happens. No place on earth where has no weather. The same should be true about your fiction. It doesn’t need to be intrusive. Just a touch once in a while will do. “The sunlight glinted on the water.” That sentence serves to tell the reader both the sun was shining and the action was by a lake. Have a character put on sunglasses, a rain coat, or even close a window against the wind.
An occasional mention, is the best way to do the job. Weather will make your action appear real. It can be a player in your story. It can keep your characters in one place. People who are snowed in are kept in place, should you require them to be trapped. Snow also makes roads slick and can cause tension through the danger. Make the character take its life in its hands to go somewhere. Simply slow them down so they — and readers — worry about actually being able to accomplish their goal, whatever that may be.
Wind and rain can do this as well. They can help you increase tension, or break it a little with some whimsical action, like an umbrella inverting. This might seem cliched, but umbrellas turn inside out in real life so an inverted umbrella can happen in your tale. Is severe weather going on outside your house? Have a storm go on in your writing as well, if such elements are appropriate. You won’t put a snowstorm in the tropics, unless under magical circumstances or something horrible happened which would cause such storms in tropical locations.
Weather also sets the mood. “It was a dark and stormy night,” is a cliche for a reason. Bright sunshine at a funeral can emphasize the grief in fiction, as in reality.
The addition of weather can make your action more realistic, as long as you prevent it from dominating the action and can also make scenes work by throwing up barriers. A hurricane can force your characters to do something they might not want to do. Snow and, to a somewhat lesser extant,rain will give them additional problems to deal with.
Weather needn’t appear in every scene, but mentioning what the weather does,gives your narration realism. That seems like I am hammering my point home, but having weather in your work means your story’s action isn’t happening in a vacuum, as simple as that.
Two characters arguing until a clap of thunder shakes the house, stopping the argument for a moment. This can create tension. The reader wonders how the squabble will resolve, that’s a way to increase tension. Weather has the capacity to break tension. Wind can blow someone’s hat or toupee off causing a problem for one character and a laugh for other characters as well as readers.
Weather doesn’t need to do anything more in your writing than prove the characters are living in the real world. The only way weather shouldn’t be in the story would be if it takes place on the Moon or in space. Weather is everywhere and should be in your story too.