How can they possible judge the correctness of the prosecution without knowing the facts of the case?
Because that's the way it works.
An appellate court either State or Federal will look at any 'new" evidence that could possibly emerge if it has the potential to change the outcome. They would then overturn that conviction and order a new trial. But that is very rare.
Outside of that, they will only overturn a conviction and send it back to a trial court if it can be demonstrated that either the judge, jury or prosecutor did not do their jobs correctly and according to the rules.
They don't judge innocence or guilt. They only review the process used to reach that verdict. See this from the ABA. http://www.abanet.org/publiced/courts/appeals.html