Basically there are three parts.

そう　・　させて　・　いただきます

そう in this case simply means 'that' or, more specifically, 'as you previously said.' Ex. 映画に行きましょうか？　いいね、そうしましょう。　(Do you want to go to a movie? Yeah, lets do that.)

させて is the causative form of する, which in Japanese depending on the context can either be 'make do' or 'let do.' Because of the nature of the conversation as well as the ending (いただきます) this case is fairly obviously 'let do.'

いただきます in this case is not the いただきます you say when you begin eating, but rather the verb いただく which is the humble (polite) form of もらう.

In other words, the phrase させていただきます has a slight implication that you are imposing a burden upon the person you are talking to and that you are grateful and apologetic to them.

As you mentioned in your post, this is an excellent way to confirm permission. For example: どうぞ、座ってください　(Please sit down.) どうも、そうさせていただきます。　(Thank you, I will do so.)