(Am I the only one who sometimes runs up against G2G's 8,000-character limit? Oh. Don't answer that. I guess I am. But I wrote this, ain't gonna delete it, so will break it into two parts.)
I agree with RJ but might state it slightly differently. DNA is a form of evidence just like any other form of genealogical evidence. As such, any given piece of evidence fits somewhere along a sliding scale ranging from "useless; discard" to "prima facie; very strong." To apply the genealogical proof standard, we create hypotheses about our tree, and then weigh and evaluate the evidence.
DNA is no different. It's about the most exciting thing since sliced bread for genealogists, but far, far too many believe that, because it's "science," the results are always going to be binary, either a "yes" or a "no." Doesn't work that way. Biology is all random and squishy to start with, and our current technological state of DNA testing and interpreting requires an amount of assumptive math, modeling, and probabilities that I think would shock most genealogists. Where RJ said "cautious approach" I might substitute "careful approach." Your hypotheses don't necessarily have to be overly-cautious, but if you want the outcome to be correct and not some confabulated addition to the many thousands of fantasy trees floating around the Internet, you have to be stringent in the evaluation of the evidence and strict in the application of the genealogical proof standard.
And, quite frankly, too many genealogists are trying to evaluate DNA evidence without ever taking the time to learn enough about the science and the math to be qualified to do so. It ain't rocket brain surgery, but it's a very, very different--learned and studied--skill set than is traditional white-glove genealogical research. Mistaken conclusions are being drawn every day, left and right, outside of WikiTree as well as within. And, like the fantasy trees that flooded the Internet confusing so many of our collective genealogies, one mistaken assumption about DNA is made, and then someone else comes along and takes that as "scientific proof" of the relationship and away we go again. If I had a dollar for every DNA determination I've seen in the past year that had weak, questionable, unsupportable, or even almost certainly false assumptions behind it, well...
Lindis, your question is an absolutely great one, and it's the correct one. It's fundamental to this whole arena. Because it's fundamental, though, there's no quick answer to it...at least not beyond Peter's irrefutable statement that shared segments come from shared ancestors. :-) The not-always-simple trick then becomes identifying what really is the shared segment; determining if the segment is valid or at least likely to be valid; and deciding if and how it can be used in your DNA matching.
If I had to concoct some quick advice for dealing with autosomal DNA (mind you, the non-recombinant forms yDNA and mtDNA have to be treated much differently), it would be--after self-education--to start high and work low. Start with yourself and the most immediate family you have. Does it add immediately to your genealogy? Nope. But it does set necessary baselines and help give you hands-on practice with available tools. At GEDmatch, do a quick "Are your parents related?" pass. If this shows your parents share any significant segments of DNA, it means you're dealing with pedigree collapse and it changes the hypotheses you apply to all other matches you find.
If you have grandparents or parents alive, test them...immediately and in that order. Then test siblings and half-siblings. Few of us get the luxury of having both parents tested but, boy, if you do it's a game changer; you can approach the DNA matching thing with a whole 'nother level of available tools and confidence. Even having one parent tested is a big deal.