This standard of proof means that, if we accuse someone of a crime ("Jimmy Jones Stole the Cookies From the Cookie Jar") and that person denies the claim (Mr. Jones says "Not me, Couldn't be!") that there are two stories that are in conflict, one must be true. Once I make the charge and present my evidence, I must show that my accusation is true. Jones on the other hand, needs to prove that it could be false.
Assuming a search of both the crime scene and the suspects property show evidence of access to the scene by prints on the front door and Jones' fingerprints on the lid, but none of the stash of missing cookies in Mr. Jones' property but we do find a key to the crime scene, we can show that Jones was at the crime scene... but that's because he's a good friend and I gave him the key to my place. So, we can assume he didn't steal the key.
Jones also points out that the police did find a partial print which matched neither myself nor Mr. Jones and I confirm that we are the only too people with keys to the house, this means that someone was inside the house who had not been there. Mr. Jones also points out that there is no finding of prints on the back door... but that's because my local CSI team was too underfunded to afford CSI of the level of Horatio Caine from Miami... so we got a guy who completely forgot to dust the back door for prints.
At this point, there is a reasonable doubt to Jimmy Jones stealing the missing cookies, as he did not have them, he was trusted by me and thus has little reason to steal from me... and there is an unidentified third person who was found in my home and I did not give anyone that access.
Because there is some evidence against my claim, the jury finds Jimmy Jones Not Guilty. It's important to note that legally this doesn't mean he didn't do it, but rather, I don't have any way of showing ONLY he didn't do it. Jimmy Jones' need not make a case of unidentified drug dealers who remain uncaught... only that my evidence does not fully refute his claim that there exists a possible second story.
In short, the Prosecution, once at trial, must show that their claims have enough evidence so that 12 random people (typical Jury rules... there are some states, where it's as low as 10 out of 12, but most are 12) can agree that this is the only possible way this could happen (100% probable). Where as the Defense, needs to show that there is the possibility for some probability of a second story. The Prosecution cannot be doubted at all.
There's any number of cases in real life cases that show that because of evidence. In real life, the OJ Simpson case went for this, some evidence of bias on the parts of the investigators, and some improper evidence storage (the famous gloves) that called the case into doubt. Also the crap treatment of the jury by both sides.
The real life Casey Anthony case is as close to a best example of a Real Life modern Rashamon as can be seen in a tried case.
The fictional events of the aforementioned Rashamon deal with the idea of who do you believe when the seemingly impossible course to reconcile events play out (Namely Three people all claim to be solely guilty to the same murder and their description of the events surrounding their guilt... including the murder victim!).
"My Cousin Vinny" is a great story about how objective material evidence is better than eyewitness testimony. That just because you are falsely accused of a crime, it does not mean that the prosecution's side is evil (almost all of the the typical "bad guys" in a legal drama are shown to be people who are doing their jobs and made honest, but hilarious mistakes). The definitive blow to the prosecution's case is a sight to behold... and the fact that everything is foreshadowed earlier makes it great on rewatch.
Twelve Angry Men shows how Jury Deliberations are really decided by 12 different perspectives having to agree on the conviction and how difficult it is to get a vote in either direction.
And then there is To Kill a Mockingbird which inspired untold numbers of American Trial Lawyers (especially defense lawyers) because of the obvious abuse of the system being permitted by the community writ large.