I think it is like in football when you are on your opponents side of the field it means you are closer to getting a touchdown than your opponent is.
Space
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when you have more space it is easier to manuever your pieces. likewise having less space means it i harder to manuever your pieces, in other words you may feel cramped.
The general rule of thumb is that the side with more space should attempt to keep pieces on the board, since they can probably place all their pieces on sensible squares, while the side with less space may try to trade off pieces, essentially leaving more room per each one of their pieces and alleviating the cramp slightly.
Usually space is determinded by your pawns.Very much what NM-Dale said, your pawns are kind of like a fench around your property, the area of the board you own.The more board you own/the more advanced your pawns, the more space you should have.
The final frontier.
With the opening that you use, it takes skill to gain space. Space makes it easier to attack, defend, and manuever.
But remember, having too much space, is going to many holes in your own side, thus a white pawn on e5, d5, f4 might not be so good, if your opponet can have enough time to maneuver inside your camp. The best way to use a space advantage is to try to make some weakness, or even attack. It's like development, if you have more development, and just make normal moves, and so. Then suddenly you don't have that advantage anymore, and you might be left with an isolated pawn.
What is it and how do I use it to my advantage? Every once in awhile I'll see something about a spatial advantage or someone having more space. Is it a tactic?