The term "bridge" refers to spanning the gap between two (seemingly) distinct market segments. Apparently, camera manufacturers at some point figured out that there were people who wanted more than the entry level/amateur camera (typically a compact with auto modes as
@Pieter12 suggest), but weren't willing to pay (financially, in terms of weight, complexity etc) for the 'pro' option (SLR or mirrorless with exchangeable lens, lots of features etc). The manufacturers identified a gap between these two segments and decided to bridge that with a product that has desirable features of both categories.
If you're cynical, you could consider a "bridge camera" as a "stuck in the middle" camera. Or "neither fish, flesh nor fowl", if you will. Of course, I'm not a cynic, so...