I'm thinking of making a mobile game but was curious on the high level idea with implementing in-app purchase.

My assumption is, correct me if I'm wrong, that you'd have some button to buy something in your game and when pressed it makes some kind of call to iTunes/GooglePlay to charge a certain value, and then comes back to your app if everything worked and then we would handle the changes to the game settings to show this purchase?

Is this grossly understated or in the right ballpark?

At a very high level, yes that is correct.

There is more to the process, of course, such as setting up your account with Google or Apple, provisioning the product, marking them as consumed, and so on. The exact details depend on which system you are using.

The documentation for Android is located here, for your reference. Apple has their own similar system you can look up.