Myth: Spiders come into houses in the fall to get out of the cold.
This seemingly simple idea conceals many false assumptions. In reality, house spiders are usually not the same species
as the yard or garden spiders outside the house.
House spiders belong to a small number of species specially adapted for indoor conditions (constant climate, poor food supply, very poor water supply). Some house spider species have been living indoors at least since the days of the Roman Empire, and are seldom to be found outside, even in their native countries (usually Europe). Many of these species now live in houses worldwide, and most have been carried by commerce to more than one continent. Few are adapted to North American outdoor environments.
House spiders colonize new houses by egg sacs carried on furniture, building materials and so forth. They usually spend their entire life cycle in, on or under their native building. If a large number appear at a specific season, it is usually late summer (August and September) -- not a notably cold time of year! -- rather than fall, and their appearance coincides with the mating season of the given species. What you are seeing is sexually mature males wandering in search of mates.
The females and young remain hidden for the most part, in crawlspaces, storage areas and other neglected rooms; wall and floor voids; behind furniture and appliances, etc. Generally fewer than 5% of the spiders you see indoors have ever been outdoors.
In contrast, outdoor spider species are not adapted to indoor conditions. Any North American spider that needed artificial shelter for the winter, would have been extinct long before Europeans arrived! Spiders are "cold-blooded" and not attracted to warmth. They don't shiver or get uncomfortable when it's cold, they just become less active and eventually, dormant. Most temperate zone spiders have enough "antifreeze" in their bodies that they won't freeze at any temperature down to -5° C.; some can get colder. The few typical outdoor spiders that do end up indoors, die or at least don't reproduce.
Myth: "I'm very kind to spiders; when I find one in the house, I put it back outside instead of killing it."
You can't put something "back" outside that was never outside in the first place. Although some house spider species can survive outdoors, most don't do well there, and some (which are native to other climates) will perish rather quickly when removed from the protective indoor habitat. You're not doing them a favor.
In any case, house spiders are mostly harmless and beneficial. Human property rights mean nothing to other species. There was spider habitat for millions of years where your home is now. My advice is, "just wave as they go by."