The story of the Farmer's Almanac is actually fairly mundane, and relates to the murder trial of Duff Armstrong. In Joseph Barrett's "Life of Abraham Lincoln," Barrett describes his version as "from the pen of one who professes to write from personal knowledge," and states that "the narration is believed to be substantially accurate." However, James Gridley's "Lincoln's Defense of Duff Armstrong: The Story of the Trial and the Celebrated Almanac" (Journal of the Illinois State Historical Society, April 1910) is somewhat more comprehensive and focuses purely on the Armstrong trial.
Armstrong found himself on trial for the 1857 murder of James Metzker. His father (Jack Armstrong) died shortly after Armstrong was arrested, and his father's wife (Hannah Armstrong) hired Lincoln. Sources describe Lincoln as a friend of either Jack or Hannah Armstrong, them having taken pity on him when he was much younger. The specific details of the murder and the intervening proceedings aren't particularly important, but if you're interested in them Gridley's article is fairly thorough.
The trial began on May 7th of 1858, Lincoln having arrived the day earlier. There was one main witness for the prosecution - Charles Allen - who testified that despite being quite a distance away he saw Armstrong strike Metzker with a slung-shot (a sap-like device). John Brady, a juror from the trial, described Allen as testifying that "the reason he could see a slung-shot that Armstrong had in his hand, with which he struck Metzker, was that the moon was shining very bright, about where the sun would be, at one o'clock in the afternoon." (Gridley pp. 18-19) Allen's testimony was contradicted by a number of defense witnesses, all of whom swore that Armstrong had only punched Metzker, but according to Brady the jury was more convinced by Allen's testimony.
Having had Allen repeat his testimony about the moon a number of times, Lincoln produced a farmer's almanac. Like most, it would have included information about the sunrise/sunset, the moon's cycle, and so on: here, it indicated that that the moon was right on the horizon at the date and time of the murder. Because Allen had testified specifically and repeatedly that the moon was overhead, and that it had allowed him to see Armstrong's weapon, this destroyed the credibility of his testimony. Armstrong was acquitted.
I can't say all that much about the validity of the various sources/accounts, but as someone on the legal side of things I can say that it's a believable account. It's very common to challenge an eyewitness's credibility based on their (in)ability to see what they claimed to have seen, or where their description conflicts with some fact that you can firmly establish. Here, it was both.
As for the almanac, it was not being used as "evidence" per se. Rather, Lincoln would have used it as a source of information under the doctrine of judicial notice. The doctrine is a long-standing one, and allows a court to "take notice" of certain facts which are either so widely or authoritatively known that they aren't reasonably in doubt. For example, the modern US Federal Rules of Evidence allow a court to take notice of things "generally known within the territorial jurisdiction of the trial court," or which are "capable of accurate and ready determination by resort to sources whose accuracy cannot reasonably be questioned."
The position of the sun or moon aren't really matters of debate, and things like sunrise/sunset or the day of the week on a given calendar date are quite common matters of judicial notice. Thus, while a farmer's almanac would not have been a broadly accepted source of evidence, what matters is that everyone present accepted it was probably correct about the position of the moon, and that the position of the moon at a given time wasn't an appropriate topic for them to debate as a live issue in court.
