I haven't the slightest idea why (inner or outer) semi-direct group products are an interesting construction. I understand inner direct products, because you're just giving conditions for which a group can be considered the direct product of two of its subgroups, and I "get" direct products. They're a very simple construction, and showing that a group decomposes into that structure is a very strong statement.
But the outer semi-direct product construction seems totally arbitrary and bizarre. What's the intuition, here?