Ok, so the basic premise of the comic is that Rumsiel accidentally misfiled Ash into a filing cabinet marked 'girl' and pushed a couple of Em's papers under a rug. This had the effect of changing Ash's gender, but nothing about his underlying personality, and de-aging Em by two years, complete with loss of work done in that time.
Now, the first premise of this explaination is that The Filing System (TFS) is a metaphor for Great Cosmic Works that can be grokked by mere mortals. This means that I can re-metaphor it to be a computer filing system with little trouble. This will help it make a little more sense. (Also, Heaven apparently doesn't use Revisioning Control or a journaled filing system. Then again, this was probably written *way* in the dark ages before people thought of it. Heck, from what I've seen, it's basically running on VMS. ;-)
Ok, we'll start with Em. She's simpler to explain, and I can use it to explain a couple of things about how TFS works. So, there is a folder marked "Emily MacArthur". Inside of it, there is a table of contents that explains which page (or 'file') has on it. Each file is basically a list that get parsed a certain way dependant on where the folder is stored, so that the same string in that list can be either a giraffes neck length, or the thickness of a whales blubber. Also, the filing system has apparently infinite processing power and I/O bandwidth. So, when he moved those two files out of the folder, and then refiled it, TFS re-read the file, noted that there were references to things that, according to Emily MacAurthur's file, hadn't happened, and so commented them out (or deleted the references to, I'm not sure which) in the rest of the universe. This means that she woke up having been born two years later in this universe. 
However, her index shows that they do, in fact, exist, and probably some minor references to them (harvard, driving, etc). Hence, she can remember that these things should exist, but she can't remember *how*, which explains why she doesn't want to take calculus again. She remembers that it was hard, but probably not specifically what was hard about it. (maybe this gets stored somewhere else?).
As for her driving 'being old hat', there are two plausible explainations. Explaination #1 (and the one that makes more sense) is that she was taught well while she had her permit, and also knew that she could do it, so had confidence, and so didn't tense up, etc. Explanation #2 is that muscle memory and 'skills' are stored in a different file then 'knowledge'. This makes sense in a way, because amniesiacs will sometimes maintain skills that they had before (talking, tying shoes, martial arts, etc), without knowledge (passwords, algebra, their name).  . As for why she was still friends with Molly, remember that their parents set that friendship up. This still would have happened, except that, instead of being equals, Em is now in a lower class, and so is an inferior (although, without seeing their relationship dynamic in the pre-misfile universe, who knows how it changed. But it's implied that they were more buddy buddy).
Now, on to the much more complex discussion of Ash. Remember how the files work? You have a set of files inside of a folder, inside of a cabinet (we'll call that a 'volume' from now on.) So, Ash Upton's file has a list of attributes that parse out, while filed in the 'human, male' volume to a 5'11", size 10 shoe wearing, 17 year old virgin boy, with certain primary and secondary sexual attributes. However, when filed in the "Human, Female" volume, she winds up 5'6", size 6 show wearing female with drastically different primary and secondary sexual characteristics.
Simple enough, right? So far, we're not too far off from Em's problem. Except that Ash's retcon is much more drastic then being the same person, just two years younger. Ash's retcon means that Ash's interactions with everybody, ever, were different. Even though Ash has the same base persona female as male, there are certain major differences. Notably, actually sending the letter to Mom, and sleeping with Jake, the best friend. The first one is, while interesting from a character and story perspective, not that interesting from a mechanics point of view, so we'll ignore it. However, the second one is mechanically much more interesting. Ash currently has expressed romantic interest in Em, because Ash currently has the memories and persona of a male expressed, heterosexual Ash. However, the retcon, female expressed Ash which our current expression of Ash doesn't have the memories of, *also* expresses as heterosexual, meaning that while RetconFemaleAsh was still best friends with Jake, she was also willing to express that friendship in a sexual manner. Something that CurrentFemaleAsh can understand logically, but is unable to come to grips with (having probably never questioned his sexuality, sleeping with the person who he thinks of as his male buddy.
Footnotes: (A note on footnotes. You'll notice that they're not nessecarilly in chronological order. They actually are, but chronologically by time that I thought about them.)
 See About:Filing System
 Actually, this might mean that TFS does, in fact, do Revision Control, but it does it by checking everything into it's own branch, and never checking things into a trunk. This is expressed with parallel universes. But this discussion is, while interesting, beyond the scope of this document.
 See #155
 This also gets into a deeper discussion of whether math is a skill or knowledge, which is, once again, beyond the scope of this document.
 Also, I'd say that the fact that Em that she taught herself how to do math by the sort of brute force method of storing how to do calculus in her head. The entire thing. Without it being broken down into bits that she didn't have to think about.
 It's said explicitly that he never had a girlfriend, but I can't remember where. I'll dig it up later if it becomes important.
 These are made up figures used for comparison only, except for the female height.
 Unlike the write up of Em, when discussing Ash, I'll probably not be using pronouns too much. They get too complex, and english doesn't handle gender swap in multiple timeframes when the primary doesn't think of themselves as their current gender expression. And if you think that last sentence was complicated, you can see what I mean.
 This is the biggest indicator that The Universe is an infinite number of branches. The data just got mispopulated to the wrong folder, so that the history files got mixed. There is also a universe where female Ash is a boy now, and having their own set of problems.