Reddit mentions of Statistics: The Art and Science of Learning from Data (3rd Edition)

I TFed an intro undergrad course that used Alan Agresti's Statistical Methods for the Social Sciences. I didn't read much of it, but the students seemed to like it. He also has another book that's probably also pretty good. The intro course for non-stats students at my graduate school is Applied Statistics for the Behavioral Sciences, which might also be worth a look. If those are too technical or hands-on, then the "for Dummies" book might also be a good choice - it's in very plain language and tries to keep things relevant to real-life examples.

Many of the bigger-picture "whys" become more apparent when you have a solid grounding in probability theory and the theory behind statistical inference, though. Some of them don't have very satisfying answers, either (Q: Why p = 0.05? A: Convention). In my opinion, the more you understand statistics, the more you realize it's less about finding exact answers than it is about quantifying imprecision. That can be hard for a layperson to wrap their head around!