Okay, let's see if I can explain it this way...
First, part of your confusion (in this, and in a lot of other topics in this thread) comes from your insistence on declaring that things must be 100% A or 100% B. The living world is not so black and white. The range of living things is a continuum more often than it's either/or. And not just across time, either -- several people have asked you to ponder the existence of "ring species", but I haven't seen you tackle it yet.
Furthermore, creationists often fail to appreciate the significance of the "nested hierarchies" of living things. It's as incorrect to say that a specific creature must be *either* a human *or* an ape as it is to say that a creature must be *either* a lion *or* a cat. Ponder that one for a moment, and then you'll be ready to understand the point of the essay You Are an Ape. Please read it.
Finally, even if you cling to the view that there's some "required" combination of genetic differences which, as soon as they're acquired, turn a "mere ape" into a "human", *bang*, that still doesn't make the evolution of one into the other a problem, or create any "breeding impossibilities". Here's how it works...
First, keep in mind that even if the "special" combination of genes which make primate DNA be considered human DNA has to all be present before *you'd* finally agree to label the resulting organism "finally human", a creature with only, say, 99% of those genes would still look pretty darned human and not so "classicly" apelike, since it would consist of 99% of the things that "separate" humans from apes. It'd only be missing one little thing out of the full set, so only one part of it would still be "apish" -- for example maybe it'd have more of a protruding brow than most people but all other human characterstics.
The other thing to keep in mind is that any one (or five, or fifty, or...) genetic differences is usually not enough to prevent interbreeding. The genetic differences just "mix and match" in members of the popuation, in the same way that both the blue-eyed gene and the brown-eyed gene swirl through human populations without any big deal.
So now that you've got some of the background, the way in which an "ape" population would evolve into a "human" population is straightforward. At some time a mutation X1 appears in the birth of a member of the population which offers some small advantage by virtue of being a small improvement (which in this example happens to bring the individual slightly closer to the advantages of being "humanlike"). The change is likely to be barely noticeable to those around him, perhaps he stands just slightly more upright, or has a slightly larger brain, or his hands are just a bit more talented, or he can voice a slightly wider range of sounds -- whatever. It's due to a small DNA change within him which just happens, by luck, to make a biochemical improvement to a particular protein in his body in a way that makes some function in his body perform just a touch better than was possible without the change. So, unlike many other mutations in the population, which made no difference, or the ones which caused damage to the functioning of the affected individual and got weeded out by natural selection, the individual who was lucky enough to receive X1 does a little better than the others in his species, and passes on his new X1 gene when he has children.
But wait, you ask, he's a "mutant", wouldn't that prevent him from mating with all the rest of the population since they don't have X1? No, it wouldn't, any more than your brown-eyed gene would prevent you from having children with a blue-eyed man. The "owner" of X1 mates with a woman who has the original form of the gene, call it Q1. Due to ordinary genetics, each of their children will have 2 X1's, or 2 Q1's, or 1 X1 and 1 Q1, by random chance. But because X1 gives a survival boost, more of the children who drew X1's from the genetic deck will have their own children than those who missed out. And so on and so on across generations, causing X1 to become more and more prevalent in the population than the competing "obsolete" Q1. Statistically, eventually X1 will "fix" in the population by virtue of being the only variety of that gene existing in the population, the Q1's having gone extinct when the last few individuals who still had a Q1 either didn't manage to have children, or had children but their children drew X1's from their parents genetic "deck".
So now the whole population is made of individuals with X1 genes and no Q1 genes.
Repeat this process for X2, another gene change which is a step along the road from "apeness" to "humanness". Then for X3, and X4, and... Finally, at some point the population will have genes X1 through X(N-1) out of the N genes which you believe are required to make them "fully human". They already look and behave pretty much entirely human, since they have almost every genetic feature which makes a species human, but you're still unwilling to declare them human because they're missing X(N), the last gene of the set. Okay, fine -- repeat the process I described above about X1 to gene mutation X(N). The first individual which gets that mutation is now "fully human" in your book. Hooray for him. However, he really isn't noticeably different from the other members of his species, since he only varies from them by a single genetic difference. So other than being the guy (or girl) who loses that last tiny remnant of "apeness" which is barely even noticeable in the population (maybe jaws on average protrude just 3% more than his or his offspring will), he has no problem having children with the mate of his choice, because they only differ by a single mutation. And eventually his X(N) gene spreads through the population over the next fifty generations until the old-style Q(N) gene gets replaced by it, and all of his kind are now 100% human instead of 99.9% human as they had been before the X(N) mutation.
And note that all the above is *standard* population genetics, *extremely* well established as ordinary processes which occur all the time in nature. It's not just an "imagine if" story.
Also note that I've simplified it somewhat by implying that, for example, mutation X46 wouldn't happen until mutation X45 had finished "fixing" in the population. Instead, it's just as easy for it to occur and be spreading into the population *while* X45 is in the process of doing so as well, for example. But this just makes the process even *more* likely, not less. There are always multiple sets of alleles floating around in populations without ill effect -- if there weren't we'd all be identical and homozygous clones.
Frankly, though, I don't think we're fully human *yet* -- if nothing else, we really need to get rid of the ape genes we still carry that cause these damned wisdom teeth which fit nicely and were useful in the longer ape jaw but just get jammed up and cause health problems in the rear of our smaller more human jaw. It looks as if we're still waiting for X(N) and haven't quite gotten the "full human" transformation finished just yet...