Most, if not all, horn tweeters work the same way as normal tweeters. A current passing through a wire loop suspended in a magnetic field causes motion in a solid diaphram which excites the air to make sound.
Most, if not all, horn tweeters are a normal tweeter with a horn in front of it. (This may be different with some professional loudspeakers... I'm talking about consumer audio here.) A horn just carries the sound forward in an efficient way, and in some designs, helps it disperse a certain way. It's the same as cupping your hands around your mouth when you yell at somebody. So horn loudspeakers tend to be more efficient than non-horn loudspeakers (meaning they play louder with the same input level), but the horn must be well designed to keep the frequency response flat, and for the speaker to have other desirable characteristics such as a good sweet spot etc.