What is 'space' on a chessboard? I know, I know, this sounds like a ridiculous question, but bear with me.

As an example of my lack of understanding, let's take the advanced variation of the French defense.

 [title "Advanced variation of the French Defense"]
 [fen ""]

 1.e4 e6 2.d4 d5 3.e5


The opening books say something like "white advances their e-pawn to gain space in the center". Now, try as I might, I simply can't see where the extra space is that white has gained!

They can't use the f6 square or the e6 square as these are covered by pawns, knight and bishop. They can't use the e4 square as that is covered by black's e5 pawn. So where on earth is the extra space that white is supposed to have gained?
It's not a ridiculous question. Before discussing your precise example, let's cover some basic grounds:


Purely from an optimal play perspective: having more space translates into having more activity and thus more options for your pieces, and that's really the key point here. The more space your pawn structure provides, the more maneuvering possibilities you will have to re-route your pieces around the board to ever more optimal posts. Moreover, controlling more space also means eyeballing or covering your opponent's important squares, related to concepts such as "dark/light square control" in chess strategy. So it's really about the relative amount of space both sides control. For a trivial example, go to very basic cases, such as first moves, 1.e4: what is the point very naively speaking? Not only it activates some of your pieces, but it also potentially takes away options from black, namely, it stops black from establishing a pawn on either d5 or f5, therefore we say white is a tempo up, because it gets the first say in space control and thus partially dictates the types of subsequent positions.
As for general ways of defining it, not necessarily restricted to optimal play in chess strategy, you can e.g., define space in terms of the number of possible legal moves you have, or the number of squares that are under attack by your pieces, so on and so forth, all compared to the other side. 


Your position:

 [title "French defense example:"]
 [Fen "rnbqkbnr/ppp2ppp/4p3/3pP3/3P4/8/PPP2PPP/RNBQKBNR b KQkq - 0 3"]


In your example, there are very clear differences in terms of piece-activity thanks to the advanced white pawns on d4 and e5 in contrast to black's lesser advanced central pawns of e6 and d5:


white has activated both bishops, whereas black's c8 bishop is facing a wall. 
white has enabled both knights to be posted centrally on c3 and f3, whereas black's f6 square has been taken away, so the knight is bound to go via the e7 square, but you have both options of f3 and e2!
white's queen has a total of 6 legal possible moves at its disposal, black's queen is restricted to only 3! Specially Qf3, Qd3 and Qg4 all very useful and common moves for the queen, all of which are symmetrically impossible for black, because of the lesser space created by the central pawns (which in turn also disabled the c8 bishop in taking away the g4 square from white's queen!)
Last but not least, although both sides have committed to their central e and d pawns, black's prospect are further limited by the fact that in order to eventually create healthy space for their pieces, they're forced to undermine white's pawn chain with f6 at some point. Meaning black's further pawn development is also being limited here, but white's are not. 


Now all is not lost, because space alone doesn't dictate the outcome of the game and all still depends on how you decide to use that space, but the moral here is that having more space means having more choices, both in terms of possible moves and possible strategies to pursue.