Space is three-dimensional, but galaxies are very flat. That's because of some complicated gravity thingymagobber.

Well actually not all galaxies are flat. Just the spiral ones mostly. There are also oval galaxies, etc, which are fat ones.

Not flat - but large. A galaxy general consists of a plane of existence as it were - to go through one's space which may contain only one systems between top to bottom - you would go through the area they claim - a sort of broad if not vague territorial boundary.

There may be nothing there but its ours so to speak.