You're asking the wrong person, allow me...
The answer is that it's not a matter of having "same" or "different" genetic code. Every human being has a different, unique genetic code (that's why DNA matching works in criminal cases). But obviously we can still interbreed.
No "exact match" of DNA is required to interbreed, just "close enough".
And the short answer to your question (there are all sorts of fascinating complicating details) is that when a population (usually, an isolated *subpopulation*) of species X is evolving towards becoming species Y, the amount of genetic change per generation is small enough that each member of the population can continue to interbreed with the rest of the population, even if it has a mutation that hasn't yet spread to the rest of the population.
Over several generations its novel mutation does spread through the population and becomes ubiquitous in the population, and thus when the next novel mutation pops up in the population, everyone's already on the same "page" with respect to the last one, and the new mutation is no more hindrance to interbreeding than the last one originally was.
} Rinse, repeat, etc.
Eventually number of novel mutations in the population becomes so large that even though the population itself can still interbreed (because they all "evolved together" into species Y through genetic exchange), the population is "enough different" DNA-wise that it will no longer be able to interbreed with members of the *original* population of species X it split off from (which itself may be relatively unchanged, or evolved off in a different direction itself).
This is how one species splits into two (or more), each "daughter" species unable to mate with its "sister" species, yet always able to breed with itself at every stage along the way.
Look back a few posts for a discussion of "ring species", whereby each subgroup along a "ring" around a mountain or whatever is still able to interbreed with its "neighbor" subgroups on the ring, but when the far "arms" of the ring meet each arm has changed enough genetically that they are unable to mate at the point where they "meet up" on the other side of the geographic obstacle. This works in a way similar to my description above -- each subgroup is "not too different" from its neighbors to interbreed, but over the whole extent of the line/ring, the far "ends" have diverged enough from each other to be unable to mate. Same thing, basically.