If we don't have souls, why does it feel so very much like we do?

The thing which distinguishes a theoretical universe from a real one, the thing that makes reality real, is my qualia. I take it as axiomatic that things which I can sense are real, and go from there. Reality itself is defined by my qualia, so it doesn't make sense to explore why I have qualia by looking at reality. Asking why we feel is like asking why reality is real. It only makes sense to ask what we feel, and what is reality - and the two are synonymous.

I'm mostly content to say, "Eventually neuroscientists will piece apart enough mental processes that we can describe the neural activity that causes this sensation to arrive". I also classify this sense of a "soul" in the same as something like the colour red. Why does red look like red? I don't know. I intend to eventually find out, but I'm not sure where to start yet.

At the most optimistic, we will be able to completely predict neural activity and behavioral outputs generated from a human system given a set of inputs. By definition, there is no way to test whether or not a system is feeling a subjective sensation.

This is one of those questions (like free will, etc) that you can solve using philosophy alone. You don't need to bring science into it - although neuroscience might eventually force you to confront the problem and help you phrase the question in a way that makes sense.