The rdi register is used as a scratchpad to save the incoming rcx value for later – first it is used to restore the possibly clobered rcx value for the second child function call, and then it becomes the return value going into rax. However, since we don’t want to clobber its value for our caller, we first save its old value and then, at the very end, restore it. This is done through a traditional push/pop pair, which in x64 is only allowed in the function prologue and epilogue.
Similarly, ebx (the bottom half of rbx) is used to save the incoming value of the second parameter in edx. However, here we don’t use a push/pop pair, but instead save it in a slot in the shadow space. Why two different mechanisms for two different registers? Sorry, that’s one for the compiler writers to answer.
There’s a lot of detail involved here, but it’s well worth the read.