I'm not sure I understand your question. The rest of the comments/answers have interpreted it in one way, but my answer to the question:
Here's where I'm lost. If I'm adding a +1 to the exponent on the LHS, where would I add it to the factorial on the RHS?
Is that the RHS will look like $(k+1)!$. I'm not sure why you might have thought otherwise? Did you think it might be $k! + 1$? Remember that you are inducting on $k$. So wherever you saw a $n$ you'd replace it with $k+1$.
If this was indeed the confusion you had it would be really really worth it to make sure you understand induction better (and maybe even refresh yourself on the definition of a function?). I can write up a loose description of induction that might help you if you want it.