Controlling board space- especially central squares- provides room to maneuver pieces. It cramps the opponent’s position, who will have difficulty bringing pieces out from behind their pawns. And it enables a quick redeployment of pieces from one zone of action (queenside, for example) to another zone of action (kingside) to initiate an attack or muster a defense around the king.
Be careful not to overextend when grabbing space. In the following position, white appears to control more space than black. Black’s counterplay demonstrates, however, that white merely occupies space but does not control it.
Black moves his knight to safety (and attacks the e5 central square) then takes advantage of white’s greediness.
Fortunes have changed. Now black controls more territory.