Half-spaces are the two vertical strips of the pitch between the wide touchline channel and the central lane, roughly level with the edges of the penalty area. A player on the ball there is harder to defend than one on the touchline or in the centre, because a single defender must choose between going central or covering the width.
The term translates the German word Halbraum and spread into English-language football writing through German tactical analysis in the mid-2010s, chief among it the website Spielverlagerung, before English pundits and coaches picked it up too.
Attacks built through the half-space are hard to defend cleanly. A winger cutting inside from there, or a midfielder receiving between the lines, can shoot, pass into the box or drive at goal, and a defender who commits to closing down one option opens up the other two.
It is one reason modern wide players drift inside rather than hug the touchline, and one reason the traditional winger who only wants the ball at the byline has become rarer than it used to be.