Teams that press high ask players to repeat short, intense efforts again and again, often without the ball. The physical profile is a bit different from simply covering lots of distance.
Pressing involves frequent accelerations, decelerations and short sprints, often in quick succession
Good aerobic fitness helps players recover between pressing actions
Collective organisation matters as much as individual fitness; a well-coordinated press can reduce wasted running
Fatigue later in matches tends to reduce high-intensity actions, which can make the press less effective
In training, many coaches build pressing fitness through games with pressing triggers: for example a 6v6 plus goalkeepers where the team must win the ball back within a few seconds or concede a point. It trains the physical and tactical side together.