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My understanding is this: In the United States, in a jury trial the jury is the finder of fact, but they are not invited (or permitted) to share any of their findings beyond guilty/not guilty. I also know that a verdict of guilty can be appealed. Also, appellate courts aren't supposed to rule on questions of fact.
This, to me, looks like the court has no way of having any facts. So, what exactly do the appeals courts base their decisions on?
The appellate court gets the trial transcript which includes all the testimony (and exhibits or descriptions/photos of them) and the trial court's rulings. Also, each side presents an appeal brief, which will generally include a statement of facts, pointing out ones it thinks favorable to its views.
In civil cases a jury may be asked/instructed to give yes/no answers to a series of questions, but this is not normal in criminal cases.
An appellate court can (and sometimes does) rule that there was not sufficient evidence to permit any reasonable jury to convict; to that extent it can rule on questions of fact. But, as the comment by ohwilleke mentions, an appellate court does not hear new evidence. It can overturn the conviction and send the case back for a new trial, and in such a new trial additional evidence may be introduce at the trial court level.
The law. If Marley is convicted for shooting the sheriff, there will be some line of evidence showing that he did the deed, and facts regarding whether the shooting was justified. The law (manifested at trial as the judge's actions) allows certain evidence to be considered, or bars such consideration, and the judge says what the law requires for a conviction. The jurors then make a judgment that boils down to a finding of (non-) guilt. Either side can object to allowing / barring evidence, or to the instructions (choice, lack). The appeals court does not try to re-create the jury's reasoning, it only looks at how the law was (dis)respected. There is also the possibility of appealing on a procedural legal ground where there is no objection on the record, such as that defendant had inadequate counsel, or judicial misconduct (the judge or a juror). However, this process also includes the transcripts, so the court has access to the admitted testimony, though not the thinking of the jurors.
