I believe this page from Khan academy explains pretty decently why temperature is important to living organisms:
It comes down to biochemistry. Temperature affects the molecules that make up our bodies and do everything in it. At very low temperatures water freezes and that messes up everything in the body for obvious reasons (molecules can't travel around and react with each other in solid ice the way they can in liquid water; ice takes up more space than water so the process of water freezing will break cell structures...). At very high temperature complex proteins will tend to lose their shape which is vital to their function, meaning they become useless. And beyond those two "hard" limits, living bodies use chemical reactions to run (essentially you could argue a living body is a very large set of varied chemical reactions), those chemical reactions are usually helped around by complex molecules called enzymes, and enzymes often have certain temperature ranges they work best in. This further narrows the range of temperatures various organisms can tolerate or thrive in.
So I believe this addresses one aspect of your question - why tropical spiders might not do well in a cold climate. The enzymes in their bodies are just optimized for a higher temperature than that. Moreover, temperatures in Sweden fall below freezing and cold-resistant species tend to evolve specific adaptations for that that aren't just behavioral but also chemical. The abstract of this paper discusses different such strategies in insects (I assume spiders would be similar), which involves either chemical adaptations to avoid bodily fluids freezing, or adaptations to survive them freezing:
So native Swedish spiders probably have adaptations of this kind that tropical spiders don't, and they're not adaptations that can arise in a single individual or even a few generations.
You bring up another question however, which is why can't tropical spiders adopt behavioral adaptations, like digging in the ground or hibernating? Well, for one thing "hibernating" is as much a physiological adaptation as a behavioral one. You don't just improvise the body slowing down that way by lying down for a nap. But more to the point, it isn't clear you can just improvise novel behaviors either, especially if you're a spider. Many behaviors like "burrowing to avoid the cold" are hard-wired evolved responses that organisms that haven't evolved them are going to do spontaneously. I don't know if the burrowing behavior you describe in spiders specifically is of that type. This paper makes the point that while spiders (like other arthropods) are traditionally thought of as having completely hard-wired behavior, more recent thinking shows they do display more flexible behavior as well:
Still, arthropods do show more such instinctive, hard-wired behavior than, say, mammals do. So while it's not impossible that some such spiders would happen to burrow in a way that avoids the cold (that's another point - they'd have to burrow at the right time, in the right places, in the right way... things that might not be obvious), and if they didn't have huge physiological barriers to surviving the cold even while buried, and enough survived this way to reproduce and multiply and weren't outcompeted by native spiders who weren't freezing their cuticles off this whole time... Then I suppose they could spread, but that's a lot of "ifs" and I think a tiny minority of introductions of tropical species would satisfy them, if any.