Good information above. I will speculate to answer your* question by trying to assume the point of view of someone who is an admissions officer, who feels responsibility for shaping the future of medicine. Research can be a good pre-medical and/or medical activity to undertake.
1. Schools are responsible to ensure future research occurs. If no doctors were interested in research, medical progress would stagnate and come to a halt. It is of paramount importance that a sizable portion of the future doctors will undertake future research. What's more, admissions must guess as to who will, in the future, pursue research freely when there is no pressure or strong incentive to do so (as is present in admissions processes for applicants, who might not actually be that interested in research). How do you tell who will actually be doing research in fifteen years, after they've earned an MD/DO? You can't tell for sure, so you have to cast a broad net, selecting research experience as part of the criteria, to ensure some fraction of those people will pursue it in the future.
2. Lacking any research experience as a natural science major might demonstrate questionable judgement. My opinion is that becoming a doctor is something that you should never "count on". I suspect many admissions officers think the same way. Therefore, if you are a natural science major, but "counting on" going to medical school in the hopes of merely treating patients clinically one day, and therefore choose to not pursue any research at all, your choice of major makes no sense. If you have a natural science bachelor's degree, but never pursued any research, what else were you going to do? There's nothing wrong with wanting to be a teacher, be a pipette monkey, or sit in an office and look through chemical MSDS sheets for the rest of your life, but these natural science programs are designed and run by Ph.D.'s. They generally want to prepare students for research. Yes, many will go into healthcare or industry, but natural science majors are expected to do research. It just looks weird if a natural science major hasn't done ANYTHING that they could indicate on an application as, somehow, research experience.
3. Natural science majors typically do have SOMETHING they can, legitimately, call "research experience" as part of their undergraduate program, so to fail to mention anything at all could demonstrate questionable judgement, also. Usually natural science programs have classes that involve some project, at least once, that could be called research. Did you write a mega-term paper for at least one class about a special enzyme, chemical structure or reaction, or an organism? Maybe it was a review, but meta-analysis and review articles are published in research journals. Did you do fly breeding in genetics? Did you design a unique experimental protocol and hypothesis at least one time as a natural science major undergraduate? Hell, can't you SOMEHOW, among the dozens of labs you've taken, suggest that, in some way, you learned something about research first-hand?
4. Doctors must be capable of reading and digesting research materials. When you're in school, you're told what you need to know, by people who know it. You can learn all the treatments, medications, symptoms and protocols easily when it's given to you in a bullet point list. And you can even be updated on all these things with continuing education classes, which you will take as a physician. But it's important for you to be able to understand research findings because you will understand how and where to apply certain techniques, or what the evolving aspects of current treatment are. If someone just gives you a protocol, you can't just carry it out blindly. You're THE doctor, not a mid-level provider, which brings me to the next point.
5. Research promotes and demonstrates critical thinking, attention to detail, and true understanding. As students, it's easy to slip into the mode of "swallow this text/diagram/statistic, regurgitate, ace exam, become doctor." But the real world is not like that. Mid-level providers, patients, and your colleagues will come to you at times, and you won't have a medical-mommy or -grandpa who is there to tell you what the answer is to the dilemma or question they are posing. When you choose to become a doctor, the buck stops with you. Involvement in research demonstrates that you're capable of truly understanding the significance of a treatment plan because part of empirical science is not making unwarranted assumptions. This level of thinking is indispensable as a physician because you can never apply treatment completely algorithmically. (If you could, we wouldn't need doctors and computers would suffice). There may always be a factor present in the patient's history that does not present orthodoxically, and you, therefore, will have to treat somewhat uniquely. This sort of thinking is constantly exercised in research, where people live on the edge of knowledge and operate in novelty day-in and day-out. These skills and ways of thinking are indispensable for physicians in many areas of medicine to promote good patient outcomes.
6. Some areas of medicine require life-long and constant familiarity with research and/or research findings. If you're an oncologist, and your patient has a 50% chance of surviving for six months based on current available treatments, you can't wait for that continuing education class next year to see what the hottest new chemotherapies available are. Well, you could, and indeed, you're not superhuman and capable of knowing every new available treatment all the time, but you'd be a better doctor if you were at least capable of keeping constantly abreast of emerging therapies and treatments. A lot of that stuff will come out in research articles, or be discussed among your research colleagues, who might be able to point you in the direction of new therapies to try. You could save someone's life by being familiar with new research, or involved in research.
That was fun to write! I hope this makes sense and answers your question as to why some medical schools place some importance on research.