Let’s say that I have completed the needed work on a rental property, and I get the reappraisal and refinance completed. How exactly do I go about getting my initial investment back? During the refinance portion do they just cut me a check for the newly acquired equity in my house? This is one step that seems confusing to me.

Yes, when you do a refinance and you take cash out. When you close on your refinance (just like closing on a house purchase) you will receive the funds in check or direct deposit form. Pretty easy. It sounds odd, but that is how it works, tax-free money.  You created that equity from your work, you get the rewards.