Well to explain in more detail, Elliquiy 'runs on' code written in php, which is called an interpreted language. So the script might include a lot of raw text looking like:
$displayname = '<span class="username">'.$context['user']['name'].'</span>';
Which basically just places your username (presumably) inside a set of html tags. This is understandable by a human, of course, but a computer needs to jump through some hoops to make it out, translating the various operators and values there into what are called opcodes - what the code eventually becomes when it finally gets run.
Namely, there are two string constant references (the span tags in between the quotes), two memory location references (the $ sign denotes a variable in php... I'll spare you the discussion about pointers, arrays and maps), two string concatenation operations (the periods), and an assignment operation (the equals sign).
I don't know how this opcode cache works specifically. If it actually converted it to the raw machine code, it would define the strings with pointers, create a function for the string concatenation and a pointer for that function, and return the result of the second one as the assignment. I could type up the x86 assembly for this but it would not be very helpful and I'd probably make a mistake somewhere anyway : )
So any call to that specific line after it's been cached would then know it only needs to crab that one variable, the rest of it it's already 'compiled'.
Opcode caches normally allow you to do even more than that, you could actually store your username for retrieval from memory later on, presuming it has a means of remembering you. Which is a lot faster than reading it from the database.
Which is where APC failed - there are sixteen separate instances of php running for elliquiy right now. The opcode cache was running independently for each one, so if you accessed the same one twice in a row (roughly a one in sixteen chance) - awesome! Otherwise, the variable it retrieves is not necessarily going to be the right one.
...and I hope that didn't confuse the issue.