When something "goes through", it completes successfully. Some examples of things that can "go through" are:
- an attempted credit card payment
- a business deal
- a formal request
"Making" a deal or a sale "go through" means that you help to complete it, or cause it to finish successfully:
I can tell that you're working really hard to make this deal go through, and I really appreciate the effort.
Another phrase which uses the same words but isn't related is "go through (an experience)":