What is a "Three-Dimensional Maneuver"?
According to what is shown in Episode 3, Three-Dimensional Maneuvering is indeed being able to move in all three axes. We, normally, are able to move two-dimensionally, meaning that we move along a horizontal plane. The gear introduces a vertical axis, making it possible for humans to move to a desired height as well.  



As to the advantages it might provide in fighting Titans, besides the ones Rinzwind says in his answer (avoid falling rubble and the like; and also, to make people move a lot faster), I guess the most evident one would be the fact that people do not need to fight at a height that only reaches a Titan's feet or legs and being able to rise enough to be able to attack them where it might do some actual damage. From the Attacks on Titan episodes I've seen so far (3 episodes), humans always go for the neck, meaning that this gear allows them to strike on more (presumably) vulnerable spots.