As in lanthas-san's initial interpretation, the writer just says that he proudly explained Inaba about an interesting system his staff thought of and implemented for him(= the writer) or the company. That sentence doesn't always have a nuance of his taking all the credit. However, he replied "グフフフフ…" to a tweet "全ては神谷さん(= his name)の手柄" after that. Thus, the context shows that he took all the credit and ~させてもらった has a nuance something like "Sorry my staff, but I proudly explained... he he" (probably this is just a joke, as already pointed out, though).
As for やりもらい動詞 or honorifics, it would be easier for understanding the meaning to drop those expressions.
e.g.
今日ウチの優秀なスタッフたちが面白いシステムを考えてそれを実装したのを俺がドヤ顔で稲葉に説明した。